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Explore the concept of local properties in graph theory and their relation to sharp thresholds, with applications in connectivity, perfect matchings, and hypergraphs. Discover the implications of Bourgain's Theorem and challenges in identifying symmetric properties with coarse thresholds. Delve into open problems like choosability and Ramsey properties.
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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...
