Induced restricted Ramsey theorems for spaces

AbstractThe induced restricted versions of the vector space Ramsey theorem and of the Graham-Rothschild parameter set theorem are proved

A separator theorem for hypergraphs and a CSP-SAT algorithm

We show that for every r≥2 there exists ϵr>0 such that any r-uniform hypergraph with m edges and maximum vertex degree o(m−−√) contains a set of at most (12−ϵr)m edges the removal of which breaks the hypergraph into connected components with at most m/2 edges. We use this to give an algorithm running in time d(1−ϵr)m that decides satisfiability of m-variable (d,k)-CSPs in which every variable appears in at most r constraints, where ϵr depends only on r and k∈o(m−−√). Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable (2,k)-CSPs with variable frequency r can be refuted in tree-like resolution in size 2(1−ϵr)m. Furthermore for Tseitin formulas on graphs with degree at most k (which are (2,k)-CSPs) we give a deterministic algorithm finding such a refutation

Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs

General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph H, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.Comment: 15 page

Quasirandom permutations are characterized by 4-point densities

For permutations π and τ of lengths |π|≤|τ| , let t(π,τ) be the probability that the restriction of τ to a random |π| -point set is (order) isomorphic to π . We show that every sequence {τj} of permutations such that |τj|→∞ and t(π,τj)→1/4! for every 4-point permutation π is quasirandom (that is, t(π,τj)→1/|π|! for every π ). This answers a question posed by Graham

The critical window for the classical Ramsey-Tur\'an problem

The first application of Szemer\'edi's powerful regularity method was the following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.Comment: 34 page