A combinatorial problem arising in group theory

Abstract

We consider a combinatorial problem occurring naturally in a group theoretical setting and provide a constructive solution in a special case. More precisely, in 1999 the author established a logarithmic bound for the derived length of the quotient of a finite solvable group modulo the second Fitting subgroup in terms of the number of irreducible character degrees of the group. Along the way, in two key lemmas an inductive process was used which at its core required a solution of some weak form of the combinatorial problem studied in this paper. This problem can be stated and studied without any group theoretical background, and in this paper we present the problem, discuss what is known and what the main conjecture is, and solve the conjecture in the smallest open case.