On a combinatorial problem of P. Erdös and L. Lovász

On a Problem of P . Erdös and S. Stein

is called a covering system if every integer satisfies at least one of the congruences (1) . An old conjecture of P . Erdös states that for every integer a there is a covering system with n l = c. Selfridge and others settled this question for c < 8 . The general case is still unsettled and seems difficult . A system (1) is called disjoint if every integer satisfies at most one of the congruenc...

On a problem of Erdös and Graham

In this paper we provide bounds for the size of the solutions of the Diophantine equation x(x+ 1)(x+ 2)(x+ 3)(x+ k)(x+ k+ 1)(x+ k+ 2)(x+ k+ 3) = y, where 4 ≤ k ∈ N is a parameter. We also determine all integral solutions for 1 ≤ k ≤ 10.

A clone-theoretic formulation of the Erdös-Faber-Lovász conjecture

The Erdős–Faber–Lovász conjecture states that if a graph G is the union of n cliques of size n no two of which share more than one vertex, then χ(G) = n. We provide a formulation of this conjecture in terms of maximal partial clones of partial operations on a set.

Fractional aspects of the Erdös-Faber-Lovász Conjecture

The Erdős-Faber-Lovász conjecture is the statement that every graph that is the union of n cliques of size n intersecting pairwise in at most one vertex has chromatic number n. Kahn and Seymour proved a fractional version of this conjecture, where the chromatic number is replaced by the fractional chromatic number. In this note we investigate similar fractional relaxations of the Erdős-Faber-Lo...

On a Problem in Combinatorial Geometry P . Erd ®