2025/06/06 by Péter Frankl, Jian Wang, Frankl, Peter +1
Mathematics · #Limits and Structures in Graph Theory #Analytic Number Theory Research #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2506.05704
Let n>2r>0 be integers. We consider families F of subsets of an n-element set, in which the union of any two members has size at most 2r. One of our results states that for n≥ 6r the number of members of size exceeding r in F is at most \binomn-2r-1. Another result shows that for n>3.5r the number of sets of size at least r is at most \binomnr. Both bounds are best possible and the latter sharpens the classical Katona Theorem. Similar results are proved for the odd case of the Katona Theorem as well.