Paul Erdős

Ramsey-type theorems

Abstract In this paper we will consider Ramsey-type problems for finite graphs, r -partitions and hypergraphs. All these problems ask for the existence of large homogeneous (monochromatic) configurations of a certain kind under the condition that the size of the underlying set is large. As it is quite common in Ramsey theory, most of our results are not sharp and almost all of them lead to new problems which seem to be difficult. The problems we treat are only loosely connected. So we will state and explain them section by section.

On a property of families of sets

1. Introduction. In this paper we are going to generalize a problem solved by MILLER in his paper [l] and prove several results concerning this new problem and some related questions. We mention here that some of our theorems (Theorems 8 and 10) have the interesting consequence that the topological product of Ks l-compact spaces (Lindelof spaces) is not necessarily

THE SIZE RAMSEY NUMBER

Let denote the class of all graphsG which satisfyG→(G1,G2). As a way of measuring minimality for members of, we define thesize Ramsey number ř(G1,G2) by.We then investigate various questions concerned with the asymptotic behaviour ofř.

Radius, diameter, and minimum degree

Abstract We give asymptotically sharp upper bounds for the maximum diameter and radius of (i) a connected graph, (ii) a connected trangle-free graph, (iii) a connected C 4 -free graph with n vertices and with minimum degree δ, where n tends to infinity. Some conjectures for K r -free graphs are also stated.

Covering the cliques of a graph with vertices

Abstract The folloeing problem is investigated. Given an undirected graph G , determine the smallest cardinality of a vertex set that meets all complete subgraphs K ⊂ G maximal under inclusion.

Some old and new problems in various branches of combinatorics

During my very long life I published very many papers which consist almost entirely of open problems in various branches of combinatorial. mathematics (i .e. graph theory, conbinatorial number theory, combinatorial geometry and combinatorial analysis). First of ail I give a list (with I hope few (or no) omissions) : 1 .

The construction of certain graphs

Denote by G(n) a graph of n vertices and by G(n ; m) a graph of n vertices and in edges . I(G) denotes the cardinal number of the largest independent set of vertices (i .e ., the largest set x il , . . ., xr,, r = I(G) of vertices of G no two of which are joined by an edge) . v(x), the valency of the vertex x, denotes the number of edges incident to x, c, . . . will denote positive absolute constants .

Crossing families

Jinos Pach2~4, Leonard J. Schulman3 Given n points in the plane, a crossing family is a collection of line segments, each joining two of the points, such that any two line segments intersect internally. We show that any n points in general position possess a crossing family of size at least ~, and describe an O(n log n)-time algorithm for finding one.

Rainbow Subgraphs in Edge-Colorings of Complete Graphs

Abstract We raise the following problem. Let F be a given graph with e edges. Consider the edge colorings of Kn (n large) with e colors, such that every vertex has degree at least d in each color (d

On two additive problems

Denote by A a set of x different natural numbers. The following two results are obtained: 1. (1) For sufficiently large x and A ⊂ [1, 3x − 3] there exists a subset B ⊂ A such that for some l ∈ Z ∑aiϵB ai=2l 2. (2) For sufficiently large x and A ⊂ [1, 4x − 4] there exists a subset B ⊂ A such that Σai ∈ B ai is a square-free number.