KruskalKatonaType Problems via the Entropy Method
Abstract
In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type KruskalKatonatype problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a $3$edgecolored graph with $R$ red, $G$ green, $B$ blue edges, the number of rainbow triangles is at most $\sqrt{2RGB}$, which is sharp. Second, we give a generalization of the KruskalKatona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.
 Publication:

arXiv eprints
 Pub Date:
 July 2023
 DOI:
 10.48550/arXiv.2307.15379
 arXiv:
 arXiv:2307.15379
 Bibcode:
 2023arXiv230715379C
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Information Theory;
 05D05;
 05D40;
 94A17
 EPrint:
 19 pages