Chvátal, Rödl, Szemerédi and Trotter [1] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. We prove that the same holds for 3-uniform hypergraphs. The main new tool which we prove and use is an embedding lemma for 3-uniform hypergraphs of bounded maximum degree into suitable 3-uniform ‘pseudo-random’ hypergraphs. keywords: hypergraphs; regularity lemm...

We prove a regularity lemma with respect to arbitrary Keisler measures μ on V , ν on W where the bipartite graph (V,W,R) is definable in a saturated structure M̄ and the formula R(x, y) is stable. The proof is rather quick, making use of local stability theory. The special case where (V,W,R) is pseudofinite, μ, ν are the counting measures, and M̄ is suitably chosen (for example a nonstandard mode...

In this note we discuss several combinatorial problems that can be addressed by the Regularity Method for hypergraphs. Based on recent results of Nagle, Schacht and the authors, we give here solutions to these problems. In particular, we prove the following: Let K (k) t be the complete kuniform hypergraph on t vertices and suppose an n-vertex k-uniform hypergraph H contains only o(n) copies of ...

We investigate decompositions of a graph into a small number of low diameter subgraphs. Let P (n, 2, d) be the smallest k such that every graph G = (V, E) on n vertices has an edge partition E = E0 ∪ E1 ∪ . . . ∪ Ek such that |E0| ≤ 2n and for all 1 ≤ i ≤ k the diameter of the subgraph spanned by Ei is at most d. Using Szemerédi’s regularity lemma, Polcyn and Ruciński showed that P (n, 2, 4) is...

We prove a variant of the abstract probabilistic version of Szemerédi’s regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in Lp for any p > 1. Our approach is based on martingale difference sequences.

We present a regularity lemma for Boolean functions f : {−1, 1}n → {−1, 1} based on noisy influence, a measure of how locally correlated f is with each input bit. We provide an application of the regularity lemma to weaken the conditions on the Majority is Stablest Theorem. We also prove a “homogenized” version stating that there is a set of input bits so that most restrictions of f on those bi...

We revisit regularity of SLE trace, for all κ 6= 8, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia–Rodemich–Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index min(1 + κ/8, 2), improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Vik...

Green [Geometric and Functional Analysis 15 (2005), 340–376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.

    M. and M.   Abstract: Existence and uniqueness of solution for singular 2-D systems depends on regularity condition. Simple regularity implies regularity and under this assumption, the generalized wave model (GWM) is introduced to cast singular 2-D system of equations as a family of non-singular 1-D models with variable structure.These index dependent models, along with a set of boundary co...