Constructing Ramsey Graphs Constructing Ramsey Graphs from from Boolean Function Boolean Function Representations Representations. Parikshit Gopalan Parikshit Gopalan Georgia Institute of Technology Georgia Institute of Technology Atlanta, Georgia, USA. Atlanta, Georgia, USA.
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Parikshit Gopalan Georgia Institute of Technology Atlanta, Georgia, USA.
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Thm: Degree d OR representation gives (G), (G) · nd.
Proof by the linear algebra method [Babai-Frankl].
Plugging in d = O(√n) gives a bound of 2√n.
Lower degree ) better graphs.
Outline of This Talk
I Ramsey graphs from OR representations.• New view of OR representations.• Sample constructions.• Ramsey graphs construction.
II Limitations to Symmetric Constructions.
Limitations to Symmetric Constructions
Thm : (√n) lower bound for symmetric polynomials.
For any OR representation, deg(P) £ deg(Q) = (n).Symmetry vs asymmetry question applies to Ramsey graph constructions.
P mod p, Q mod q. [BBR, Alon]Gives a representation of OR over Zpq.
Known lower bound: √(n/pq).When n < pq [Alon] …
Xi represents OR mod pq.
Both polynomials mod pa. [FW] Based on interpolation algorithm mod pa [G’06].
Thm : (√n) lower bound for symmetric polynomials.
Limitations to Symmetric Constructions
Partition Problem
Adversary gets number n. Picks1. Primes p and q where p¢q > n.2. A µ {1,…, p-1} and B µ {1, …, q-1}Every x 2 {1, …, n} is covered by A or B.Minimize |A|¢|B|.
x mod p lies in A
Partition Problem
Adversary gets number n. Picks1. Primes p and q where p¢q > n.2. A µ {1,…, p-1} and B µ {1, …, q-1}Every x 2 {1, …, n} is covered by A or B.Minimize |A|¢|B|.
1 2 3 4
1 2 3 4 5 6
p = 5, q = 7, n = 12
1 12
…
Partition LemmaTrivial Solutions :
A = {1,…, p-1} and B = {p, 2p, …, }
A = {q, 2q, …} and B = {1, …, q-1}
Gives |A|¢ |B| = n.
Better solutions ) Better OR representations.
Partition Lemma: In any solution, |A|¢|B| ¸ n/8.
Symmetry vs. Asymmetry
Do low degree OR polynomials exist?
Conjecture [Barrington-Beigel-Rudich]: No! (for representations mod 6)• Symmetric polynomials for Symmetric functions. • CRT. Hard explicit construction problem ?
Symmetric polynomials give graphs on {0,1}n based on distances.Q : Are such graphs not good Ramsey graphs?