Addressing a question of Gowers, we determine the order of the tower height for the partition size in a version of Szemerédi’s regularity lemma.

<p style='text-indent:20px;'>This paper deals with a distributed optimal control problem to the coupled chemotaxis-fluid models. We first explore global-in-time existence and uniqueness of strong solution. Then, we define cost functional establish Lagrange multipliers. Finally, derive some extra regularity for multiplier.</p>

We prove a version of Szemerédi’s regularity lemma for subsets of a typical random set in F p . As an application, a result on the distribution of three-term arithmetic progressions in sparse sets is discussed.

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...