Let H be a k-uniform hypergraph whose vertices are the integers 1, . . . , N . We say that H contains a monotone path of length n if there are x 1 < x 2 < · · · < x n+k−1 so that H contains all n edges of the form {x i , x i+1 , . . . , x i+k−1 }. Let N k (q, n) be the smallest integer N so that every q-coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic monotone path of length n. While the study of N k (q, n) for specific values of k and q goes back (implicitly) to the seminal 1935 paper of Erdős and Szekeres, the problem of bounding N k (q, n) for arbitrary k and q was studied by Fox, Pach, Sudakov and Suk.Our main contribution here is a novel approach for bounding the Ramsey-type numbers N k (q, n), based on establishing a surprisingly tight connection between them and the enumerative problem of counting high-dimensional integer partitions. Some of the concrete results we obtain using this approach are the following:• We show that for every fixed q we have N 3 (q, n) = 2 Θ(n q−1 ) , thus resolving an open problem raised by Fox et al.• We show that for every k ≥ 3, N k (2, n) = 2where the height of the tower is k − 2, thus resolving an open problem raised by Eliáš and Matoušek.• We give a new pigeonhole proof of the Erdős-Szekeres Theorem on cups-vs-caps, similar to Seidenberg's proof of the Erdős-Szekeres Lemma on increasing/decreasing subsequences.
Let W n (p, q) denote the minimum number of edges in an n × n bipartite graph G on vertex sets X, Y that satisfies the following condition; one can add the edges between X and Y that do not belong to G one after the other so that whenever a new edge is added, a new copy of K p,q is created. The problem of bounding W n (p, q), and its natural hypergraph generalization, was introduced by Balogh, Bollobás, Morris and Riordan. Their main result, specialized to graphs, used algebraic methods to determine W n (1, q).Our main results in this paper give exact bounds for W n (p, q), its hypergraph analogue, as well as for a new variant of Bollobás's Two Families Theorem. In particular, we completely determineOur proof applies a reduction to a multi-partite version of the Two Families Theorem obtained by Alon. While the reduction is combinatorial, the main idea behind it is algebraic.
A celebrated result of Gowers states that for every > 0 there is a graph G such that every -regular partition of G (in the sense of Szemerédi's regularity lemma) has order given by a tower of exponents of height polynomial in 1/ . In this note we give a new proof of this result that uses a construction and proof of correctness that are significantly simpler and shorter.
We introduce a new variant of Szemerédi's regularity lemma which we call the sparse regular approximation lemma (SRAL). The input to this lemma is a graph G of edge density p and parameters ǫ, δ, where we think of δ as a constant. The goal is to construct an ǫ-regular partition of G while having the freedom to add/remove up to δ|E(G)| edges. As we show here, this weaker variant of the regularity lemma already suffices for proving the graph removal lemma and the hypergraph regularity lemma, which are two of the main applications of the (standard) regularity lemma. This of course raises the following question: can one obtain quantitative bounds for SRAL that are significantly better than those associated with the regularity lemma?Our first result answers the above question affirmatively by proving an upper bound for SRAL given by a tower of height O(log 1/p). This allows us to reprove Fox's upper bound for the graph removal lemma. Our second result is a matching lower bound for SRAL showing that a tower of height Ω(log 1/p) is unavoidable. We in fact prove a more general multicolored lower bound which is essential for proving lower bounds for the hypergraph regularity lemma.
Constructing a spanning tree of a graph is one of the most basic tasks in graph theory.Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph G consisting of (1+ )n edges (for a prespecified constant > 0), where the decision for different edges should be consistent with the same subgraph G . Can this task be performed by inspecting only a constant number of edges in G? Our main results are:• We show that if every t-vertex subgraph of G has expansion 1/(log t) 1+o(1) then one can (deterministically) construct a sparse spanning subgraph G of G using few inspections. To this end we analyze a "local" version of a famous minimum-weight spanning tree algorithm.• We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3-regular graphs of high girth, in which every t-vertex subgraph has expansion 1/(log t) 1−o(1) . We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth.
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