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...