The solution and applications of a combinatorial problem

In this paper, we solve the following combinatorial problem. Let A(1), A(2), ... ,A(p) be given sets and B-1, B-2, ... ,B-q be m-sets. We lower bound the number q of sets B-1, B-2, ... ,Bq such that boolean OR(p)(i=1) A(i) subset of boolean OR(q)(i=1) B-i, and, for each i is an element of {1, 2, ... , q}, the set B-i does not contain two distinct elements x and y with x is an element of A(j), y is an element of A(k) and j not equal k. Our result directly implies the theorems proved by Bessy et al. [S. Bessy, N. Lichiardopol, J.-S. Sereni, Two proofs of the Bermond-Thomassen conjecture for tournaments with bounded minimum in-degree, Discrete Math. 310 (C) (2010) 557-560]. 2012 Elsevier B.V. All rights reserved.