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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”.). approximable.
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Hunting for Sharp Thresholds Ehud Friedgut Hebrew University
Local properties A graph property will be called localif 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”.)
approximable by a local property. Almost- Theorem: If a monotone graph property has a coarse threshold then it is local. Non-
Applications • Connectivity • Perfect matchings in graphs • 3-SAT hypergraphs Assume, by way of contradiction, coarseness.
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 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
Semi-sharp sharp . Rules of thumb: • If it don’t look local - then it ain’t. • 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: Generalproperties • with a coarse threshold have • positive correlation with local properties. What about the common generalization? Probably true...
