Hunting for Sharp Thresholds Ehud Friedgut Hebrew University
Dec 21, 2015
Hunting for Sharp Thresholds
Ehud FriedgutHebrew University
Local properties
A graph property will be called local if it is the property of containing a subgraph from a given finite list of finite graphs.
(e.g. “Containing a triangle or acycle of length 17).”
Theorem: If a monotone graph property has a coarse threshold then it is local .
Non-
approximableby a local property.
Almost-
Applications
• Connectivity
• Perfect matchings in graphs
• 3-SAT
Assume, by way of contradiction, coarseness.
hypergraphs
Generalization to signed hypergraphs
Use Bourgain’s Theorem. Or, as verified by Hatami and Molloy:
Replace G(n,p) by F(n,p), a random 3-sat
formula, M by a formula of fixed sizeetc.; (The proof of the original criterionfor coarseness goes through.)
Initial parameters
• It’s easy to see that 1/100n < p < 100/n
• M itself must be satisfiable• Assume, for concreteness, that M
involves 5 variables x1,x2,x3,x4,x5 and that setting them all to equal “true” satisfies M.
Restrictive sets of variables
We will say a quintuple of variables {x1,x2,x3,x4,x5} is restrictive if setting them all to “true” renders F unsatisfiable.
Our assumptions imply that at least a (1-α)-proportion of the quintuples are
restrictive.
Erdős-Stone-Simonovits
The hypergraph of restrictive quintuples is super-saturated : there exists a constant β such that if one chooses 5 triplets they form a complete 5-partite system of restrictive quintuplets with probability at least β.
Placing clauses of the form ( x1 V x2 V x3)on all 5 triplets in such a system renders
F unsatisfiable!
Punchline
Adding 5 clauses to F make it unsatisfiable with probability at least
β2{-15}, so adding εn3p clauses does this w.h.p., and not with probability less than 1-2α.
Contradiction!
Applications
• Connectivity
• Perfect matchings in hypergraphs
• 3-SAT
Rules of thumb:
•If it don’t look local - then it ain’t.
•Semi-sharp sharp .
•No non-convergent oscillations.
A semi-random sample of open problems:
•Choosability (list coloring number)
•Ramsey properties of random sets of integers
•Vanishing homotopy groupof a random 2-dimensionalsimplicial complex.
A more theoretical open problem:
•F: Symmetric properties witha coarse threshold have high correlation with local properties.
•Bourgain: General propertieswith a coarse threshold have positive correlation with local properties.What about the common generalization?
Probably true...