A collection of open problems in celebration of Imre Leader's 60th birthday

One of the great pleasures of working with Imre Leader is to experience his infectious delight on encountering a compelling combinatorial problem. This collection of open problems in combinatorics has been put together by a subset of his former PhD students and students-of-students for the occasion of his 60th birthday. All of the contributors have been influenced (directly or indirectly) by Imre: his personality, enthusiasm and his approach to mathematics. The problems included cover many of the areas of combinatorial mathematics that Imre is most associated with: including extremal problems on graphs, set systems and permutations, and Ramsey theory. This is a personal selection of problems which we find intriguing and deserving of being better known. It is not intended to be systematic, or to consist of the most significant or difficult questions in any area. Rather, our main aim is to celebrate Imre and his mathematics and to hope that these problems will make him smile. We also hope this collection will be a useful resource for researchers in combinatorics and will stimulate some enjoyable collaborations and beautiful mathematics

Extremal theory of ordered graphs

We call simple graphs with a linear order on the vertices ordered graphs. Turán-type extremal graph theory naturally extends to ordered graphs. This is a survey on the ongoing research in the extremal theory of ordered graphs with an emphasis on open problems

Endre Szemerédi, Premi Abel 2012

Aquest article presenta una breu descripció de les contribucions matemàtiques més destacades d'Endre Szemerédi, Premi Abel 2012.This article presents a short description of the main mathematical contributions of Endre Szemerédi, Abel Prize 2012

Bounded VC-Dimension Implies the Schur-Erd?s Conjecture

In 1916, Schur introduced the Ramsey number r(3;m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph K_n, there is a monochromatic copy of K?. He showed that r(3;m) ? O(m!), and a simple construction demonstrates that r(3;m) ? 2^?(m). An old conjecture of Erd?s states that r(3;m) = 2^?(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension

On Ramsey Theory and Slow Bootstrap Percolation

This dissertation concerns two sets of problems in extremal combinatorics. The major part, Chapters 1 to 4, is about Ramsey-type problems for cycles. The shorter second part, Chapter 5, is about a problem in bootstrap percolation. Next, we describe each topic more precisely. Given three graphs G, L1 and L2, we say that G arrows (L1, L2) and write G → (L1, L2), if for every edge-coloring of G by two colors, say 1 and 2, there exists a color i whose color class contains Li as a subgraph. The classical problem in Ramsey theory is the case where G, L1 and L2 are complete graphs; in this case the question is how large the order of G must be (in terms of the orders of L1 andL2) to guarantee that G → (L1, L2). Recently there has been much interest in the case where L1 and L2 are cycles and G is a graph whose minimum degree is large. In the past decade, numerous results have been proved about those problems. We will continue this work and prove two conjectures that have been left open. Our main weapon is Szemeredi\u27s Regularity Lemma.Our second topic is about a rather unusual aspect of the fast expanding theory of bootstrap percolation. Bootstrap percolation on a graph G with parameter r is a cellular automaton modeling the spread of an infection: starting with a set A0, cointained in V(G), of initially infected vertices, define a nested sequence of sets, A0 ⊆ A1 ⊆. . . ⊆ V(G), by the update rule that At+1, the set of vertices infected at time t + 1, is obtained from At by adding to it all vertices with at least r neighbors in At. The initial set A0 percolates if At = V(G) for some t. The minimal such t is the time it takes for A0 to percolate. We prove results about the maximum percolation time on the two-dimensional grid with parameter r = 2

Ramsey numbers for set-colorings

For s, t, n ∈ N with s ≥ t, an (s, t)-coloring of K$_n$ is an edge coloring of Kn in which each edge is assigned a set of t colors from {1, . . . , s}. For k ∈ N, a monochromatic K$_k$ is a set of k vertices S such that for some color i ∈ [s], i ∈ c(uv) for all distinct u, v ∈ S. As in the case of the classical Ramsey number, we are interested in the least positive integer n = R$_{s,t}$(k) such that for any (s, t)-coloring of K$_n$, there exists a monochromatic K$_k$. We estimate upper and lower bounds for general cases and calculate close bounds for some small cases of R$_{s,t}$(k)