Hunting for Sharp Thresholds Ehud Friedgut Hebrew 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 a cycle of length 17”.)

Theorem: If a monotone graph property has a coarse threshold then it is local. Non- approximable by 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 size etc.; (The proof of the original criterion for 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 x 1,x 2,x 3,x 4,x 5 and that setting them all to equal “true” satisfies M.

Restrictive sets of variables We will say a quintuple of variables {x 1,x 2,x 3,x 4,x 5 } 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 ( x 1 V x 2 V x 3 ) 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 εn 3 p 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 group of a random 2-dimensional simplicial complex.

A more theoretical open problem: F: Symmetric properties with a coarse threshold have high correlation with local properties. Bourgain: General properties with a coarse threshold have positive correlation with local properties. What about the common generalization? Probably true...