. . . . . . . . . Classification problems of good distance sets and classification problems of good families Masashi Shinohara(Shiga University) Osaka Combinatorics Seminar April 16, 2016 Masashi Shinohara(Shiga University) Classification problems of distance sets and families
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. . . . . .
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. ..
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Classification problems of good distance setsand
classification problems of good families
Masashi Shinohara(Shiga University)
Osaka Combinatorics SeminarApril 16, 2016
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Contents
Connections of two topics
Distance sets
DefinitionsConvex planar distance sets (survey)Planar distance sets (survey, S.)Three-distance sets in R3 (S.)Two-distance sets in Rd (Nozaki-S.)Distance sets on circles (Momihara-S.)
Families of sets
Erdos-Ko-Rado’s theorem and Katona’s proof (survey)Some generalizations of Erdos-Ko-Rado’s theorem (survey)Union families (Frankl-Tokushige)r -wise union families (Frankl-S.-Tokushige)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. s-code
s-code in Sd−1 s-code in Fn2
X ⊂ Sd−1 is s-code if C ⊂ Fn2 is s-code if
p · q ≤ s for ∀p,q ∈ X (p = q) dH(x, y) ≥ s for ∀x, y ∈ C (x = y)
p · q: usual inner product. dH(x, y) := |{i | xi = yi}|
max{|X | | X ⊂ Sd−1 is an s-code}? max{|C | | C ⊂ Fn2 is an s-code}?
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. s-code in Fn2 and Intersecting families
[n] = {1, 2, . . . , n}2[n] := {F | F ⊂ [n]}([n]k
):= {F ∈ 2[n] | |F | = k}
For x ∈ Fn2, we define x ∈ 2n by
x := {i | xi = 1}. (support of x)
1 2 3 4 5
x 1 1 0 0 1
y 1 0 1 0 0
x ∨ y = (1, 1, 1, 0, 1), where(x ∨ y)i := max{xi , yi}
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Contents
Connections of two topics
Distance sets
DefinitionsConvex planar distance sets (survey)Planar distance sets (survey, S.)Three-distance sets in R3 (S.)Two-distance sets in Rd (Nozaki-S.)Distance sets on circles (Momihara-S.)
Families of sets
Erdos-Ko-Rado’s theorem and Katona’s proof (survey)Some generalizations of Erdos-Ko-Rado’s theorem (survey)Union families (Frankl-Tokushige)r -wise union families (Frankl-S.-Tokushige)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Finiteness of two-distance sets
There are infinitely many (d + 1)-point two-distance sets in Rd .
.Theorem (Einhorn-Schoenberg, 1966)..
.
. ..
.
.
There are finitely many (d + 2)-point two-distance sets in Rd .
For any simple graph G (= Kn,Nn) of order n,∃!X ⊂ Rd such that G ↔ X and d ≤ n − 2. (function view)
X : n-point two-distance set in Rd .
d \ n 3 4 5 6 7
2 ∞ 6 1 ×3 − ∞ 26 6 ×
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Finiteness of two-distance sets
There are infinitely many (d + 1)-point two-distance sets in Rd ..Theorem (Einhorn-Schoenberg, 1966)..
.
. ..
.
.
There are finitely many (d + 2)-point two-distance sets in Rd .
For any simple graph G (= Kn,Nn) of order n,∃!X ⊂ Rd such that G ↔ X and d ≤ n − 2. (function view)
X : n-point two-distance set in Rd .
d \ n 3 4 5 6 7
2 ∞ 6 1 ×3 − ∞ 26 6 ×
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Finiteness of two-distance sets
There are infinitely many (d + 1)-point two-distance sets in Rd ..Theorem (Einhorn-Schoenberg, 1966)..
.
. ..
.
.
There are finitely many (d + 2)-point two-distance sets in Rd .
For any simple graph G (= Kn,Nn) of order n,∃!X ⊂ Rd such that G ↔ X and d ≤ n − 2. (function view)
X : n-point two-distance set in Rd .
d \ n 3 4 5 6 7
2 ∞ 6 1 ×3 − ∞ 26 6 ×
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Finiteness of two-distance sets
There are infinitely many (d + 1)-point two-distance sets in Rd ..Theorem (Einhorn-Schoenberg, 1966)..
.
. ..
.
.
There are finitely many (d + 2)-point two-distance sets in Rd .
For any simple graph G (= Kn,Nn) of order n,∃!X ⊂ Rd such that G ↔ X and d ≤ n − 2. (function view)
X : n-point two-distance set in Rd .
d \ n 3 4 5 6 7
2 ∞ 6 1 ×3 − ∞ 26 6 ×
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Three-distance sets in R2, R3
.Theorem (S, 2004)..
.
. ..
.
.
There are exactly 34 three-distance sets with 5 points.
g3(2) = 7 and optimal three-distance sets in R2 is R7 and R+6 .
.Conjecture (Einhorn-Schoenberg, 1966)..
.
. ..
.
.
Every 12-point three-distance set is isomorphic to the vertex set ofa regular icosahedron. In particular, g3(3) = 12.
.Theorem (S.)..
.
. ..
.
.
Every 12-point three-distance set is isomorphic to the vertex set ofa regular icosahedron. In particular, g3(3) = 12.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Three-distance sets in R2, R3
.Theorem (S, 2004)..
.
. ..
.
.
There are exactly 34 three-distance sets with 5 points.
g3(2) = 7 and optimal three-distance sets in R2 is R7 and R+6 .
.Conjecture (Einhorn-Schoenberg, 1966)..
.
. ..
.
.
Every 12-point three-distance set is isomorphic to the vertex set ofa regular icosahedron. In particular, g3(3) = 12.
.Theorem (S.)..
.
. ..
.
.
Every 12-point three-distance set is isomorphic to the vertex set ofa regular icosahedron. In particular, g3(3) = 12.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Three-distance sets in R2, R3
.Theorem (S, 2004)..
.
. ..
.
.
There are exactly 34 three-distance sets with 5 points.
g3(2) = 7 and optimal three-distance sets in R2 is R7 and R+6 .
.Conjecture (Einhorn-Schoenberg, 1966)..
.
. ..
.
.
Every 12-point three-distance set is isomorphic to the vertex set ofa regular icosahedron. In particular, g3(3) = 12.
.Theorem (S.)..
.
. ..
.
.
Every 12-point three-distance set is isomorphic to the vertex set ofa regular icosahedron. In particular, g3(3) = 12.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. A relation between their results and conjecture
.Proposition..
.
. ..
.
.
Every 12-point three-distance set containing a 5-point two-distanceset is isomorphic to the vertex set of a regular icosahedron.
.Proposition..
.
. ..
.
.
Every 14-point three-distance set in R3 contains a 5-pointtwo-distance set in R3.
.Proposition..
.
. ..
.
.
Every 12-point three-distance set in R3 contains a 5-pointtwo-distance set in R3.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. A relation between their results and conjecture
.Proposition..
.
. ..
.
.
Every 12-point three-distance set containing a 5-point two-distanceset is isomorphic to the vertex set of a regular icosahedron.
.Proposition..
.
. ..
.
.
Every 14-point three-distance set in R3 contains a 5-pointtwo-distance set in R3.
.Proposition..
.
. ..
.
.
Every 12-point three-distance set in R3 contains a 5-pointtwo-distance set in R3.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. A relation between their results and conjecture
.Proposition..
.
. ..
.
.
Every 12-point three-distance set containing a 5-point two-distanceset is isomorphic to the vertex set of a regular icosahedron.
.Proposition..
.
. ..
.
.
Every 14-point three-distance set in R3 contains a 5-pointtwo-distance set in R3.
.Proposition..
.
. ..
.
.
Every 12-point three-distance set in R3 contains a 5-pointtwo-distance set in R3.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Definition of diameter graphs
.Definition..
.
. ..
.
.
D(X ) := maxA(X ) :the diameter of X
G := DG (X ) :the diameter graph of X
X
{V (G ) = X ,
For P ,Q ∈ X , P ∼ Q if PQ=D(X).
.Example.... ..
.
.
DG (R2m+1) = C2m+1, DG (R2m) = m · P2
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Independent set of diameter graph
Let X be a k-distance set.
X
⊃ Y
↕ ↕V (DG (X )) ⊃ H : an independent set
Then Y is an at most (k − 1)-distance set..Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = 12.Then α(G ) ≥ 5.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Independent set of diameter graph
Let X be a k-distance set.
X
⊃ Y
↕
↕
V (DG (X ))
⊃ H : an independent set
Then Y is an at most (k − 1)-distance set..Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = 12.Then α(G ) ≥ 5.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Independent set of diameter graph
Let X be a k-distance set.
X
⊃ Y
↕
↕
V (DG (X )) ⊃ H : an independent set
Then Y is an at most (k − 1)-distance set..Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = 12.Then α(G ) ≥ 5.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Independent set of diameter graph
Let X be a k-distance set.
X
⊃
Y
↕ ↕V (DG (X )) ⊃ H : an independent set
Then Y is an at most (k − 1)-distance set..Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = 12.Then α(G ) ≥ 5.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Independent set of diameter graph
Let X be a k-distance set.
X ⊃ Y
↕ ↕V (DG (X )) ⊃ H : an independent set
Then Y is an at most (k − 1)-distance set..Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = 12.Then α(G ) ≥ 5.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Independent set of diameter graph
Let X be a k-distance set.
X ⊃ Y
↕ ↕V (DG (X )) ⊃ H : an independent set
Then Y is an at most (k − 1)-distance set.
.Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = 12.Then α(G ) ≥ 5.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Independent set of diameter graph
Let X be a k-distance set.
X ⊃ Y
↕ ↕V (DG (X )) ⊃ H : an independent set
Then Y is an at most (k − 1)-distance set..Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = 12.Then α(G ) ≥ 5.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Diameter graphs for X ⊂ R2
.Proposition..
.
. ..
.
.
Let G = DG (X ) for X ⊂ R2. Then(i) G contains no C2k for any k ≥ 2;(ii) if G contains C2k+1, then any two vertices in V (G ) \V (C2k+1)are not adjacent.In particular, G contains at most one cycle.
.Proof...
.
. ..
.
.
Two segments whose lengths are the diameter of X mustintersect.
isolated vertices
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Applications for planar distance sets
.Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R2 with |X | = n.If G = Cn, then we have α(G ) ≥
⌈n2
⌉where
⌈n2
⌉is the smallest
integer at least n2 .
.Application..
.
. ..
.
.
Let X be a 9-point 4-distance set in R2 and G = DG (X ).
If G = C9, then X = R9 (∵ X is a convex 4-distance set)
If G = C9, then α(G ) ≥⌈92
⌉= 5. Therefore X contains a
5-point (at most) 3-distance set in R2.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Applications for planar distance sets
.Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R2 with |X | = n.If G = Cn, then we have α(G ) ≥
⌈n2
⌉where
⌈n2
⌉is the smallest
integer at least n2 .
.Application..
.
. ..
.
.
Let X be a 9-point 4-distance set in R2 and G = DG (X ).
If G = C9, then X = R9 (∵ X is a convex 4-distance set)
If G = C9, then α(G ) ≥⌈92
⌉= 5. Therefore X contains a
5-point (at most) 3-distance set in R2.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Applications for planar distance sets
.Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R2 with |X | = n.If G = Cn, then we have α(G ) ≥
⌈n2
⌉where
⌈n2
⌉is the smallest
integer at least n2 .
.Application..
.
. ..
.
.
Let X be a 9-point 4-distance set in R2 and G = DG (X ).
If G = C9, then X = R9 (∵ X is a convex 4-distance set)
If G = C9, then α(G ) ≥⌈92
⌉= 5. Therefore X contains a
5-point (at most) 3-distance set in R2.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Diameter graphs for X ⊂ R3
.Theorem (Dol’nikov(2000))..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3. If G containstwo cycles of odd length C and C ′, then V (C ) ∩ V (C ′) = ∅.
.Corollary..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = n.Let G contain a triangle C. Then G − C is a bipartite graph. Inparticular, α(G ) ≥
⌈n−32
⌉.
.Corollary..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3. Then G doesnot contain two disjoint 5-cycles.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Diameter graphs for X ⊂ R3
.Theorem (Dol’nikov(2000))..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3. If G containstwo cycles of odd length C and C ′, then V (C ) ∩ V (C ′) = ∅.
.Corollary..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = n.Let G contain a triangle C. Then G − C is a bipartite graph. Inparticular, α(G ) ≥
⌈n−32
⌉.
.Corollary..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3. Then G doesnot contain two disjoint 5-cycles.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Diameter graphs for X ⊂ R3
.Theorem (Dol’nikov(2000))..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3. If G containstwo cycles of odd length C and C ′, then V (C ) ∩ V (C ′) = ∅.
.Corollary..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = n.Let G contain a triangle C. Then G − C is a bipartite graph. Inparticular, α(G ) ≥
⌈n−32
⌉.
.Corollary..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3. Then G doesnot contain two disjoint 5-cycles.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. α(DG (X )) ≥ 5 for X ⊂ R3 with |X | = 12
.Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = 12.Then α(G ) ≥ 5, where α(G ) is a independence number of G.
Let G = DG (X ) for X ⊂ R3 with n (≥ 12) points.By Corollary, if G contains a triangle,
α(G ) ≥⌈12− 3
2
⌉= 5.
We assume a simple graph G satisfy the following conditions.|V (G )| = 12
α(G ) < 5
triangle− free
Then we can prove that G contains disjoint 5-cycles C and C ′.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. α(DG (X )) ≥ 5 for X ⊂ R3 with |X | = 12
.Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = 12.Then α(G ) ≥ 5, where α(G ) is a independence number of G.
Let G = DG (X ) for X ⊂ R3 with n (≥ 12) points.By Corollary, if G contains a triangle,
α(G ) ≥⌈12− 3
2
⌉= 5.
We assume a simple graph G satisfy the following conditions.|V (G )| = 12
α(G ) < 5
triangle− free
Then we can prove that G contains disjoint 5-cycles C and C ′.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. α(DG (X )) ≥ 5 for X ⊂ R3 with |X | = 12
.Proposition..
.
. ..
.
.
Let G = DG (X ) be the diameter graph of X ⊂ R3 with |X | = 12.Then α(G ) ≥ 5, where α(G ) is a independence number of G.
Let G = DG (X ) for X ⊂ R3 with n (≥ 12) points.By Corollary, if G contains a triangle,
α(G ) ≥⌈12− 3
2
⌉= 5.
We assume a simple graph G satisfy the following conditions.|V (G )| = 12
α(G ) < 5
triangle− free
Then we can prove that G contains disjoint 5-cycles C and C ′.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Contents
Connections of two topics
Distance sets
DefinitionsConvex planar distance sets (survey)Planar distance sets (survey, S.)Three-distance sets in R3 (S.)Two-distance sets in Rd (Nozaki-S.)Distance sets on circles (Momihara-S.)
Families of sets
Erdos-Ko-Rado’s theorem and Katona’s proof (survey)Some generalizations of Erdos-Ko-Rado’s theorem (survey)Union families (Frankl-Tokushige)r -wise union families (Frankl-S.-Tokushige)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Example of a two-distance set in Sd−1 ⊂ Rd
g2(d) := max{|X | : X is a 2-distance set in Rd}g∗2 (d) := max{|X | : X is a 2-distance set in Sd−1}
Vd : the set of all mid-points of edges of a regular simplex in Rd .Then
|Vd | =(d + 1
2
)=
d(d + 1)
2
and Vd is a two-distance set in Rd .
.Proof...
.
. ..
.
.
Let P1 ↔ E1, P2 ↔ E2 for P1,P2 ∈ Vd . (Ei : edge)(a) E1 and E2 have a common vertex ⇒ P1P2 = 1/2(b) E1 and E2 don’t have a common vertex ⇒ P1P2 =
√2/2
d(d + 1)
2≤ g∗
2 (d) ≤ g2(d)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Example of a two-distance set in Sd−1 ⊂ Rd
g2(d) := max{|X | : X is a 2-distance set in Rd}g∗2 (d) := max{|X | : X is a 2-distance set in Sd−1}
Vd : the set of all mid-points of edges of a regular simplex in Rd .Then
|Vd | =(d + 1
2
)=
d(d + 1)
2
and Vd is a two-distance set in Rd ..Proof...
.
. ..
.
.
Let P1 ↔ E1, P2 ↔ E2 for P1,P2 ∈ Vd . (Ei : edge)(a) E1 and E2 have a common vertex ⇒ P1P2 = 1/2(b) E1 and E2 don’t have a common vertex ⇒ P1P2 =
√2/2
d(d + 1)
2≤ g∗
2 (d) ≤ g2(d)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Tight two-distance sets
Upper boundsDelsarte-Goethals-Seidel 1977
d(d + 1)
2≤ g∗
2 (d) ≤(d + 2
2
)− 1 =: T (d)
Blokhuis 1983, Bannai-Bannai-Stanton 1983
d(d + 1)
2≤ g2(d) ≤
(d + 2
2
)−1 =: T (d)
A two-distance set X in Sd−1 (resp. Rd) is said to be tight if
|X | = T (d)(resp. |X | =
(d+22
)= T (d) + 1
).
.Remark..
.
. ..
.
.
Tight two-distance sets in Sd−1 are known only ford = 2, 6, 22.
It is known some conditions for d to exist tight two-distanceset in Sd−1. (Bannai-Damerell, Bannai-Munemasa-Venkov)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Optimal two-distance sets and other results
d 1 2 3 4 5 6 7 8
g∗2 (d) 2 5 6 10 16 27 28 (36)
g2(d) 3 5 6 10 16 27 29 45
.Theorem (Musin(2008), JCTA)..
.
. ..
.
.
g∗2 (d) =
d(d+1)2 for 8 ≤ d ≤ 39 (d = 22, 23) and
g∗2 (22) = 275 = T (22), g∗
2 (23) = 276 or 277.
.Theorem (Nozaki-S. (2010), JCTA)..
.
. ..
.
.
∃ tight two-distance set in Sd−2(⊂ Rd−1) ⇐⇒∃ proper locally two-distance set in Rd with more than d(d+1)
2points ..Theorem (Nozaki-S. (2012), LAA)..
.
. ..
.
.
G: strongly regular graph of order n ⇐⇒d(G ) + d(G ) = n − 1 and both X (G ) and X (G ) are spherical.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Contents
Connections of two topics
Distance sets
DefinitionsConvex planar distance sets (survey)Planar distance sets (survey, S.)Three-distance sets in R3 (S.)Two-distance sets in Rd (Nozaki-S.)Distance sets on circles (Momihara-S.)
Families of sets
Erdos-Ko-Rado’s theorem and Katona’s proof (survey)Some generalizations of Erdos-Ko-Rado’s theorem (survey)Union families (Frankl-Tokushige)r -wise union families (Frankl-S.-Tokushige)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Distance sets on circles.Theorem (Momihara-S., to appear in AMM)..
.
. ..
.
.
Let X ⊂ S1 be a k-distance set with n points.If k < Mn, then X ⊂ R2k or X ⊂ R2k+1.
.Proposition 1..
.
. ..
.
.
Let X ⊂ S1 be a k-distance set with n points.If k < Mn, then X ⊂ Rm for some m..Proposition 2..
.
. ..
.
.
Let X ⊂ Rm be a k-distance set with n points. Assume that⟨X ⟩ = Rm. If k < Mn, then m ∈ {2k, 2k + 1}.
Differences in A ⊂ Zm.A = {0, 1, 4} ⊂ Z9.A− A = {0,±1,±3,±4}.|A− A| = 7↔ 3-distance set
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Distance sets on circles.Theorem (Momihara-S., to appear in AMM)..
.
. ..
.
.
Let X ⊂ S1 be a k-distance set with n points.If k < Mn, then X ⊂ R2k or X ⊂ R2k+1..Proposition 1..
.
. ..
.
.
Let X ⊂ S1 be a k-distance set with n points.If k < Mn, then X ⊂ Rm for some m.
.Proposition 2..
.
. ..
.
.
Let X ⊂ Rm be a k-distance set with n points. Assume that⟨X ⟩ = Rm. If k < Mn, then m ∈ {2k, 2k + 1}.
Differences in A ⊂ Zm.A = {0, 1, 4} ⊂ Z9.A− A = {0,±1,±3,±4}.|A− A| = 7↔ 3-distance set
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Distance sets on circles.Theorem (Momihara-S., to appear in AMM)..
.
. ..
.
.
Let X ⊂ S1 be a k-distance set with n points.If k < Mn, then X ⊂ R2k or X ⊂ R2k+1..Proposition 1..
.
. ..
.
.
Let X ⊂ S1 be a k-distance set with n points.If k < Mn, then X ⊂ Rm for some m..Proposition 2..
.
. ..
.
.
Let X ⊂ Rm be a k-distance set with n points. Assume that⟨X ⟩ = Rm. If k < Mn, then m ∈ {2k, 2k + 1}.
Differences in A ⊂ Zm.A = {0, 1, 4} ⊂ Z9.A− A = {0,±1,±3,±4}.|A− A| = 7↔ 3-distance set
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Distance sets on circles.Theorem (Momihara-S., to appear in AMM)..
.
. ..
.
.
Let X ⊂ S1 be a k-distance set with n points.If k < Mn, then X ⊂ R2k or X ⊂ R2k+1..Proposition 1..
.
. ..
.
.
Let X ⊂ S1 be a k-distance set with n points.If k < Mn, then X ⊂ Rm for some m..Proposition 2..
.
. ..
.
.
Let X ⊂ Rm be a k-distance set with n points. Assume that⟨X ⟩ = Rm. If k < Mn, then m ∈ {2k, 2k + 1}.
Differences in A ⊂ Zm.A = {0, 1, 4} ⊂ Z9.A− A = {0,±1,±3,±4}.|A− A| = 7↔ 3-distance set
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Sum set (cf. B. Nathanson, Additive number theory, Inverse problmems...)
.Definition..
.
. ..
.
.
G: a finite abelian group A,B: subsets of G .A+ B = {a+ b | a ∈ A, b ∈ B}.
.Example..
.
. ..
.
.
We take subsets A,B,C of Z11 where ,A = {0, 1, 2},B = {0, 2, 4},C = {3, 4, 5}.
Then
A+ B = {0, 1, 2, 3, 4, 5, 6},A+ C = {3, 4, 5, 6, 7}.
.Remark.... ..
.
.
We may assume 0 ∈ A ∩ B.
.Theorem (Cauchy(1893)-Davenport(1935))..
.
. ..
.
.
Let G = Zp (p : prime).|A+ B| ≥ min{p, |A|+ |B| − 1}.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Vosper’s theorem
.Theorem (Cauchy(1893)-Davenport(1935))..
.
. ..
.
.
Let G = Zp (p : prime).|A+ B| ≥ min{p, |A|+ |B| − 1}.
.Theorem (Vosper(1956))..
.
. ..
.
.
Let G = Zp (p : prime). If |A+ B| = |A|+ |B| − 1, then one ofthe followings hold:
|A| = 1 or |B| = 1.
|A+ B| = p − 1.
A and B are arithmetic progressions with same commondifference.
A = {a+ id | i = 0, 1, . . . , k−1},B = {b+ id | i = 0, 1, . . . , ℓ−1}.
ThenA+ B = {a+ b + id | i = 0, 1, . . . , k + ℓ− 2}.Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Kneser’s theorem.Definition..
.
. ..
.
.
G: finite abelian group,A,B ⊂ G (subset)
A+ B = {a+ b | a ∈ A, b ∈ B}..Theorem (Cauchy-Davenport)..
Let X = {a0, a1, . . . , an−1} be a k-distance set on S1.Assume that al(ai , ai + 1) ∈ Q for i = 1, 2, . . . , n − 1.If k < n − 1, then X ⊂ Rm for some m.
Since k < n − 1,∃ai , aj ∈ X such that al(a0, ai ) = al(a0, aj)
Then al(a0, a1) = al(a0, ai )− al(a1, ai )
= al(a0, aj)− al(a1, ai ) ∈ Q>0
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Rational distance sets → Circular distance set
odd even
.Proposition..
.
. ..
.
.
Let X be a k-distance set with n points on S1 with k < Mn.
If n is even, then X ⊂ Rm for some m.
If n is odd and both L(ai ) and R(ai ) are rational, thenX ⊂ Rm.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Irrational distance set → Circular distance set
.Proposition..
.
. ..
.
.
Let X be a k-distance set with n points on S1 with k < Mn.Then both R(ai ) and L(ai ) are rational.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Irrational distance set → Circular distance set
.Proposition..
.
. ..
.
.
Let X be a k-distance set with n points on S1 with k < Mn.Then both R(ai ) and L(ai ) are rational.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Irrational distance set → Circular distance set
.Proposition..
.
. ..
.
.
Let X be a k-distance set with n points on S1 with k < Mn.Then both R(ai ) and L(ai ) are rational.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Main results for distance sets on circles
Mn =
{3t, if n = 4t or 4t − 1,
3t − 2, if n = 4t − 2 or 4t − 3
.Theorem (Momihara-S. to appear in AMM)..
.
. ..
.
.
Let X ⊂ S1 be a k-distance set with n points.If k < Mn, then X ⊂ R2k or X ⊂ R2k+1.
.Problem..
.
. ..
.
.
k , d : given.
Determine gd(k).Classify k-distance sets in Rd with gd(k) points or close togd(k) points.When are there finitely many k-distance sets in Rd
d , n: given . . .
n, k: given . . .
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Contents
Connections of two topics
Distance sets
DefinitionsConvex planar distance sets (survey)Planar distance sets (survey, S.)Three-distance sets in R3 (S.)Two-distance sets in Rd (Nozaki-S.)Distance sets on circles (Momihara-S.)
Families of sets
Erdos-Ko-Rado’s theorem and Katona’s proof (survey)Some generalizations of Erdos-Ko-Rado’s theorem (survey)Union families (Frankl-Tokushige)r -wise union families (Frankl-S.-Tokushige)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Intersecting families.Definition..
.
. ..
.
.
A family F is called intersecting (resp. t-intersecting) ifF ∩ F ′ = ∅ (resp. |F ∩ F ′| ≥ t)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Katona’s proof
.Lemma..
.
. ..
.
.
Let n ≥ 2k. Let C be a cyclic permutation on [n]. If F ⊂ C (k) isintersecting, then |F| ≤ k.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Katona’s proof
By a double counting of
p(F) := |{(C ,F ) | F ∈ F ,C is a circ . perm. on [n],F ∈ C (k)}| .
|F| · (n − k)! · k! = p(F) ≤ (n − 1)! · k.
|F| ≤ (n − 1)! · k(n − k)! · k!
=(n − 1)!
(k − 1)!(n − k)!=
(n − 1
k − 1
).
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Contents
Connections of two topics
Distance sets
DefinitionsConvex planar distance sets (survey)Planar distance sets (survey, S.)Three-distance sets in R3 (S.)Two-distance sets in Rd (Nozaki-S.)Distance sets on circles (Momihara-S.)
Families of sets
Erdos-Ko-Rado’s theorem and Katona’s proof (survey)Some generalizations of Erdos-Ko-Rado’s theorem (survey)Union families (Frankl-Tokushige)r -wise union families (Frankl-S.-Tokushige)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Erdos-Ko-Rado’s theorem (general case)
.Theorem (Erdos-Ko-Rado(1961))..
.
. ..
.
.
F ⊂([n]k
): t-intersecting family
If n ≥ n0(k , t), then|F| ≤
(n − t
k − t
).
.Theorem (Frankl (1978))..
.
. ..
.
.
The best possible bound for n0(k, t) is (k − t + 1)(t + 1) fort ≥ 15.
.Theorem (Wilson (1984)).... ..
.
.
The best possible bound for n0(k, t) is (k − t + 1)(t + 1) for all t.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Intersecting families when n < (k − t + 1)(t + 1)
.Example (non-trivial t-intersecting family)..
.
. ..
.
.
Fi = {S ∈([n]
k
)| |S ∩ [t + 2i ]| ≥ t + i} (0 ≤ i ≤ ⌊(n − t)/2⌋)
is t intersecting.
1 · · · i i + 1 · · · 2i 1 + 2i · · · t + 2i · · · n
× · · · × ⃝ · · · ⃝× · · · × ⃝ · · · ⃝
.Theorem (Ahlswede-Khachatrian (1996))..
.
. ..
.
.
Let F ⊂([n]k
)be a t-intersecting. Then
|F| ≤ max0≤i≤ n−t
2
|Fi |
holds. Moreover, equality holds only for F = Fi for some i.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Other problems
t-intersecting family F ⊂([n]k
).
Non-trivial intersecting family for n > n0(k, t).(Hilton-Milner(1967), Ahlswede-Khachatrian(1996))r -wise t-intersecting family (Brace-Daykin(1971),Frankl-Tokushige(2002,2005), Tokushige(2005,2007,2010))
Other situations
Intersecting family for F ⊂ 2[n] (Katona(1964))Multisets (Furedi-Gerbner-Vizer(2014))Cross-intersecting familyJohnson graph, Hamming graph (other (Q-polynomial)distance regular graphs)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Contents
Connections of two topics
Distance sets
DefinitionsConvex planar distance sets (survey)Planar distance sets (survey, S.)Three-distance sets in R3 (S.)Two-distance sets in Rd (Nozaki-S.)Distance sets on circles (Momihara-S.)
Families of sets
Erdos-Ko-Rado’s theorem and Katona’s proof (survey)Some generalizations of Erdos-Ko-Rado’s theorem (survey)Union families (Frankl-Tokushige)r -wise union families (Frankl-S.-Tokushige)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Intersecting families and Union families
.Theorem (Erdos-Ko-Rado(1961), Frankl (1978), Wilson (1984))..
.
. ..
.
.
Given n ≥ k ≥ t > 0 and a t-intersecting family F ⊂([n]k
). If
n ≥ (k − t + 1)(t + 1), then|F| ≤
(n − t
k − t
).
.Definition..
.
. ..
.
.
A family F is called s-union if|F ∪ F ′| ≤ s
holds for all F ,F ′ ∈ F ..Remark..
.
. ..
.
.
Let F ⊂([n]k
). Since |F ∪ F ′| = 2k − |F ∩ F ′| for F ,F ′ ∈
([n]k
),
F : t-intersecting ⇐⇒ F : (2k − t)-union.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Union for q-ary codes
1 2 3 4 5
x 1 1 0 0 1
y 1 0 1 0 0
x ∨ y = (1, 1, 1, 0, 1), where(x ∨ y)i := max{xi , yi}
12
4
35
x y[5]
.Definition (join)..
.
. ..
.
.
For x, y ∈ {0, 1, . . . , q − 1}n, x ∨ y is defined by(x ∨ y)i := max{xi , yi}
(2, 3, 1) (1, 2, 2) (2, 3, 2)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Union families in Nn ({0, 1, . . . , q − 1}nfor large q])
.Definition..
.
. ..
.
.
N = {0, 1, 2, . . .}.For a = (a1, a2, . . . , an) ∈ Nn, |a| := a1 + a2 + · · ·+ an.
For a,b ∈ Nn, we define the join a ∨ b by
(a ∨ b)i := max{ai , bi}.A ⊂ Nn is s-union if
|a ∨ b| ≤ s for all a,b ∈ A.
wn(s) := max{|A| | A ⊂ Nn is s-union}.For a,b ∈ Nn, we let a ≺ b iff ai ≤ bi for all 1 ≤ i ≤ n.
a ∈ A is maximal if @ b ∈ A such that a ≺ b.
For x ∈ Nn, D(x) := {y ∈ Nn | y ≺ x} (down set)..Remark..
.
. ..
.
.
If A ⊂ Nn is s-union, then (D(A) :=)∪a∈AD(a) is also s-union.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. D(a) and balanced partition
D(x) := {y ∈ Nn | y ≺ x}
.Example (10-union in N3)..
.
. ..
.
.
Let a = (4, 3, 3). Then D(a) is 10-union with|D(a)| = (4 + 1)× (3 + 1)2 = 80. Therefore w3(10) ≥ 80.
.Definition (balanced partition)..
.
. ..
.
.
b = (b1, b2, . . . , bn) ∈ Nn is called a balanced partition iff|bi − bj | ≤ 1 for 1 ≤ i < j ≤ n.
.Lemma..
.
. ..
.
.
If b is a balanced partition and a is not a balanced partition with|b| = |a|, then
|D(a)| < |D(b)|.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Balanced partition
.Proposition..
.
. ..
.
.
wn(s) := max{|A| | A ⊂ Nn is s-union}.
w1(s) = |D(a1)| = s + 1,
w2(2s) = |D(a2)| = (s + 1)2,
w2(2s + 1) = |D(a3)| = (s + 2)(s + 1).
where ai is a balanced partition with |ai | = s, 2s, 2s + 1,respectively. i. e. a1 = (s), a2 = (s, s), a3 = (s + 1, s).
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Examples and upper set S(a, d).Example (Another example of 10-union)..
.
. ..
.
.
A := {(4, 2, 2), (2, 4, 2), (2, 2, 4), (3, 3, 3)}.Then A is 10-union. (We will check soon.) Moreover,
A = {a+ 2e1, a+ 2e2, a+ 2e3} ∪ {a+ 1},where ei is the i-the standard base of Rn, 1 :=
∑ei , a = (2, 2, 2).
.Definition (upper set at distance d from a ∈ Nn)..
.
. ..
.
.
S(a, d) = {a+ ϵ : ϵ ∈ Nn, |ϵ| = d}
D(A) = D (S(a, 2)) ∪ D (S(a+ 1, 0)) where a = (2, 2, 2).
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. D(A) = D (S(a, 2)) ∪ D (S(a+ 1, 0)) where a = (2, 2, 2)
If p,q ∈ D (S(a+ 1, 0)), then clearly |p ∨ q| ≤ 9.
If p,q ∈ D (S(a, 2)), thenp ∨ q ∈ D (S(a, 2 + 2)) .
Therefore |p ∨ q| ≤ 10.
If p ∈ D (S(a, 2)) and q ∈ D (S(a+ 1, 0)),
p ∨ q ∈ D (S(a+ 1, 2− (3− 2))) .
Therefore |p ∨ q| ≤ 10..Definition..
.
. ..
.
.
Kn(a, d) :=
⌊ dn−1
⌋∪i=0
D (S (a+ i1, d − (n − 1)i)) .
K3(a, 2) =1∪
i=0
D (S (a+ i1, 2− 2i)) where a = (2, 2, 2).
.Proposition.... ..
.
.
Kn(a, d) is (|a|+ 2d)-union.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Good construction of s-union
Kn(a, d) :=
⌊ dn−1
⌋∪i=0
D (S (a+ i1, d − (n − 1)i))
.Proposition.... ..
.
.
Kn(a, d) is (|a|+ 2d)-union..Proof...
.
. ..
.
.
Let 0 ≤ i ≤ j ≤ ⌊ dn−1⌋, and
b ∈ D (S (a+ i1, d − (n − 1)i)) ,c ∈ D (S (a+ j1, d − (n − 1)j)) .
|b \ c| : =∑
1≤l≤n
max{bl − cl , 0}
≤ d − (n − 1)i − (j − i),
|b ∨ c| = |c|+ |b \ a|≤ (|a|+ d + j) + d − (n − 2)i − j
≤ |a|+ 2d − (n − 2)i ≤ |a|+ 2d .
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. |Kn(a, d)| and balanced partition b
.Lemma..
.
. ..
.
.
|Kn(a, d)| =n∑
j=0
(d + j
j
)σn−j(a)
+
⌊ dn−1
⌋∑i=1
((d − (n − 1)i + n
n
)−
(d − (n − 1)i + n − 1
n
)).
whereσk(a) =
∑K∈([n]k )
∏i∈K
ai .
.Lemma..
.
. ..
.
.
Let |a| = |b|. If b is a balanced partition, then|Kn(a, d)| ≤ |Kn(b, d)|.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Known Fact
.Conjecture (Frankl-Tokushige, to appear in JCTA)..
.
. ..
.
.
Let n,s be given. Then it follows that
wn(s) = max0≤d≤⌊s/2⌋
|Kn(a, d)|
where a ∈ Nn is a balanced partition with |a| = s − 2d .
.Remark..
.
. ..
.
.
Frankl-Tokushige[1] verified the conjecture for the following cases:
(i) s = 3, (It is somewhat surprising that the case n = 3 is not so easy,
and the formula for w3(s) is rather involved.)
(ii) n > n0(s),
(iii) Under two suppositions;
a is well-defined{P1,P2, . . . ,Pn} ⊂ A
where a and Pi ’s are defined from an s-union family A ⊂ Nn.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Polytope Pn(a, d) with |a| = s − nd
Let n, s be given, and a = (a1, a2, . . . , an) ∈ Nn with |a| = s − 2dfor some d ∈ N. We define convex polytope P = Pn(a, d) ⊂ Rn bythe following equations:
xi ≥ 0 (1 ≤ i ≤ n),
xi ≤ ai + d (1 ≤ i ≤ n),
xi + xj ≤ ai + aj + d (1 ≤ i < j ≤ n).
L = Ln(a, d) := {x ∈ Nn : x ∈ P}.Lemma.... ..
.
.Two set K and L are the same, and s-union.
Kn(a, d) =
⌊ dn−1
⌋∪i=0
D (S (a+ i1, d − (n − 1)i))
In [1], they prove K = L by (i) K ⊂ L and (ii) |K | = |L|.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Definition of Pi ’s for s-union A(⊂ Nn)
m = m(A) := (m1,m2, . . . ,mn) where mi := max{xi : x ∈ A}
d = d(A) :=|m| − s
n − 2. (d ∈ N?)
a = a(A) := (a1, a2, . . . , an) where ai = mi − d . (a ∈ Nn?)
Then s = |a|+ 2d since d =|m| − s
n − 2and |a| = |m|+ nd .
Pi := a+ dei .
.Remark.... ..
.
.If ai < 0, then Pi is not defined. For n = 3, we have ai ≥ 0.
WLOG, we may assume that m1 ≥ m2 ≥ m3. Then a1 ≥ a2 ≥ a3.Since A is s-union, m1 +m2 ≤ s. Then |m| − s ≤ m3.d = |m| − s ≤ m3. Therefore a3 ≥ m3 − d ≥ 0.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. A ⊂ L3(a, d)
L = L3(a, d) := {x ∈ N3 : x satisfies (1), (2), (3)}xi ≥ 0 (1 ≤ i ≤ 3), (1)
xi ≤ ai + d (1 ≤ i ≤ 3), (2)
xi + xj ≤ ai + aj + d (1 ≤ i < j ≤ 3). (3)
mi := max{xi : x ∈ A}, d = |m|−s, ai := mi−d , s = |a|+2d ..Lemma..
.
. ..
.
.
Let A ⊂ N3 be s-union, and a := a(A) and d := d(A). Then
A ⊂ L3(a, d)..Proof...
.
. ..
.
.
(1) Trivial since A ⊂ N3.(2) For any x ∈ (x1, x2, x3) ∈ A, xi ≤ mi = ai + d .(3) For any x ∈ (x1, x2, x3) ∈ A, take y = (∗, ∗,m3) ∈ A.Then |a|+ 2d = s ≥ x1 + x2 +m3 = x1 + x2 + a3 + d .This implies a1 + a2 + d ≥ x1 + x2.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Summary for s-union family in Nn
.Lemma..
.
. ..
.
.
Ln(a, d) = Kn(a, d) and they are (|a|+ 2d)-union.
Let |a| = |b|. If b is a balanced partition, then|Kn(a, d)| ≤ |Kn(b, d)|.
1. If A ⊂ Ln(a, d), then
2. A ⊂ Kn(a, d) since Ln(a, d) = Kn(a, d).
3. |A| ≤ |Kn(b, d)| for a balanced partition b with |b| = |a|.
|A| ≤ |Ln(a, d)| = |Kn(a, d)| ≤ |Kn(b, d)|
|m|−sn−2 ∈ N? mi − d ≥ 0? {Pi} ⊂ A? A ⊂ L!=⇒
Yes=⇒Yes
=⇒Yes
⇓ ⇓ ⇓? |A|:small? |A|:small?No No No
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Contents
Connections of two topics
Distance sets
DefinitionsConvex planar distance sets (survey)Planar distance sets (survey, S.)Three-distance sets in R3 (S.)Two-distance sets in Rd (Nozaki-S.)Distance sets on circles (Momihara-S.)
Families of sets
Erdos-Ko-Rado’s theorem and Katona’s proof (survey)Some generalizations of Erdos-Ko-Rado’s theorem (survey)Union families (Frankl-Tokushige)r -wise union families (Frankl-S.-Tokushige)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. r -wise union families in Nn
.Definition..
.
. ..
.
.
a,b, . . . , z ∈ Nn, we define the join a ∨ b ∨ · · · ∨ z by
(a ∨ b ∨ · · · ∨ z)i := max{ai , bi , . . . , zi}.A ⊂ Nn is r -wise s-union if
|a1 ∨ · · · ∨ ar | ≤ s for all a1, a2, . . . ar ∈ A.
.Definition..
.
. ..
.
.
Kn(r , a, d) :=
⌊d/u⌋∪i=0
D (S (a+ i1, d − ui))
where u = n − r + 1.
.Conjecture..
.
. ..
.
.
Let r ≥ 2 and A be a r -wise s-union in Nn. Then
|A| ≤ max0≤d≤⌊s/r⌋
|Kn(r , a, d)|
where a ∈ Nn is a balanced partition with |a| = s − 2d .
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. r -wise s-union families in Nk for k ≤ r
.Proposition (s-union family for n ≤ 2)..
.
. ..
.
.
wn(s) := max{|A| | A ⊂ Nn is s-union}.w1(s) = |D(a1)| = s + 1,
w2(2s) = |D(a2)| = (s + 1)2,
w2(2s + 1) = |D(a3)| = (s + 2)(s + 1).
where ai is a balanced partition with |ai | = s, 2s, 2s + 1,respectively. i. e. a1 = (s), a2 = (s, s), a3 = (s + 1, s).
.Proposition..
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. ..
.
.
Let A be r-wise s-union in Nk for k ≤ r . Then
|A| ≤ |D(a)|
where a ∈ Nn is a balanced partition with |a| = s.
Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Main Theorem.Theorem (Framkl-Tokushige)..
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. ..
.
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Conjecture for s-union is true for the following cases:
(i) s = 3,
(ii) n > n0(s),
(iii) Under two suppositions;
a is well-defined{P1,P2, . . . ,Pn} ⊂ A
where a and Pi ’s are defined from A..Theorem (Frankl-S-Tokushige)..
.
. ..
.
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Conjecture for r -wise s-union is true for the following cases:(i) s = r + 1,
(ii) n > n0(r , s),
(iii) Under two suppositions;
a is well-defined{P1,P2, . . . ,Pn} ⊂ A
where a and Pi ’s are defined from A.Masashi Shinohara(Shiga University) Classification problems of distance sets and families
. . . . . .
.. Correspondence table
u := n − r + 1
(2-wise) s-union 2-wise s-union|a1 ∨ a2| ≤ s (∀a1, a2 ∈ A) |a1 ∨ · · · ∨ ar | ≤ s (∀a1, . . . , ar ∈ A)
K
⌊ dn−1 ⌋∪i=0
D (S (a+ i1, d − (n − 1)i))
⌊ du ⌋∪
i=0
D (S (a+ i1, d − ui))
Lxi ≥ 0 (1 ≤ i ≤ n),xi ≤ ai + d (1 ≤ i ≤ n),xi+xj ≤ ai+aj+d (1 ≤ i < j ≤ n)
xi ≥ 0(1 ≤ i ≤ n),∑i∈I
xi ≤∑i∈I
ai + d
(1 ≤ I ≤ n − r + 1, I ⊂ [n]),
d |m|−sn−2
|m|−sn−r
|a| s + 2d s + rd
|Kn(r , a, d)| =n∑
j=0
(d + j
j
)σn−j(a)
+
⌊d/u⌋∑i=1
n∑j=u+1
((d − ui + j
j
)−(d − ui + u
j
))σn−j(a+ i1)
Masashi Shinohara(Shiga University) Classification problems of distance sets and families