Notes on use of generalized entropies in counting

TL;DR

This paper explores the application of generalized entropies, specifically Tsallis-Havrda-Charvát entropies, to derive bounds in combinatorial counting problems, extending classical results like Shearer's lemma and Bregman's theorem.

Contribution

It introduces a novel approach using generalized entropies to obtain new bounds in counting problems, including extensions of Shearer's lemma and Bregman's theorem.

Findings

Derived upper bounds on the size of families of subsets with restricted intersections.

Extended Shearer's lemma using Tsallis-Havrda-Charvát entropies.

Produced one-parameter extensions of Bregman's theorem.

Abstract

We address an idea of applying generalized entropies in counting problems. First, we consider some entropic properties that are essential for such purposes. Using the $\alpha$-entropies of Tsallis-Havrda-Charv\'{a}t type, we derive several results connected with Shearer's lemma. In particular, we derive upper bounds on the maximum possible cardinality of a family of $k$-subsets, when no pairwise intersections of these subsets may coincide. Further, we revisit the Minc conjecture. Our approach leads to a family of one-parameter extensions of Br\'{e}gman's theorem. A utility of the obtained bounds is explicitly exemplified.