H. L. Abbott

Intersection Theorems for Systems of Sets

Let n and k be positive integers, k ⩾ 3. Denote by φ(n, k) the least positive integer such that if T is any family of more than φ(n, k) sets, each set with n elements, then some k members of T have pairwise the same intersection. In this paper we evaluate φ(2, k) for all k ⩾ 3 and obtain a new upper bound for φ(n, k) and a new lower bound for φ(n, 3).

A Turán type problem for interval graphs

Abstract We consider the following analogue of a problem of Turan for interval graphs: Let c = c ( n , m ) be the largest integer such that any interval graph with n vertices and at least m edges contains a complete subgraph on c vertices. We determine the value of c ( n, m ) explicitly.

Sparse color-critical hypergraphs

In this paper we obtain estimates for the least number of edges ann-uniformr-color-critical hypergraph of orderm may have.

On finite Δ-systems II

Abstract A Δ(k) system is a family F of k distinct sets which have pairwise the same intersection. A weak Δ(k) system is a family F of k distinct sets such that | F þ G | = t for some non-negative integer t and all F,G ϵ F , F ≠ G. In this paper we study some functions related to these Δ-systems. In particular α(n,k) = max{| F | : | F | = n ∀ F ϵ F , F ⊅ weak Δ(k)} then α(3,3) = 10, α(m + n, k) ⩾ α (m, k) α(n, k) and α (n, 3) > c5 n 2

On finite Δ-systems

Let @Dn and k be positive integers, k>=3. By an (l, n) system is meant a family of l distinct n sets A @D(k) system is a family of k distinct sets which have pairwise the same intersection Erdos and Rado showed to each n and k that there corresponds a least integer @f(n, k) such that nl>@f(n,k), every (l, n) system contains a @D(k) system. In this paper we obtain new upper bounds for @f(3, k), and new lower bounds for @f(n, 4) and @f(n, 6).

Lower bounds for certain types of van der Waerden numbers

Abstract Denote by B ( k , l ) the least integer such that, if the numbers 1, 2, 3,…, B ( k , l ) + 1 are partitioned in any way into k sets, at least one of the sets contains an arithmetic progression of l + 1 terms, together with the common difference. Some new lower bounds are derived for B ( k , l ).

Covering Squares with Squares

Abstract. For 0<x<1 denote by f(x) the length of the side of the largest open square Q that can be covered by the sequence {Qn}∈fty n=0 of closed squares, where Qn has side of length xn, and is placed so that its sides are parallel to those of Q. We obtain some information concerning f(x) and discuss some related questions.

Color-critical graphs and hypergraphs with few edges and no short cycles

Abstract We give constructions of color-critical graphs and hypergraphs with no cycles of length 5 or shorter and with relatively few edges.

On a conjecture of Gallai concerning complete subgraphs of k -critical graphs

A graph G is said to be k-critical if it has chromatic number k but every proper subgraph of G has a (k−1)-coloring. T. Gallai asked whether each k-critical graph of order n contains at most n complete subgraphs of order k − 1. This is clearly so when k = n, and it is also true when k = 3 since the only 3-critical graphs are the cycles of odd length. M. Stiebitz recently gave a positive answer to Gallais question in the case k = 4. In this paper we give an affirmative answer for all k⩾5.

Bounds for the covering number of a graph

Abstract Upper and lower bounds for the covering number of a graph are obtained. It is shown, by probabilistic methods, that there exists a large class of graphs for which the upper bound obtained is essentially best possible.