Analogues of Katona's and Milner's Theorems for two families

TL;DR

This paper extends classical combinatorial theorems to two and three families with cross s-union properties, establishing optimal bounds on their sizes and characterizing extremal cases.

Contribution

It provides new bounds for the sizes of families with cross s-union properties, including for antichains and multiple families, generalizing Katona's and Milner's theorems.

Findings

Bound: || + ||  1 + \u2211_{i=0}^s inom{n}{i}

For antichains with n  2s, the sum is at most inom{n}{1} + inom{n}{s-1}

Results extend to three families with similar bounds

Abstract

Let $n>s>0$ be integers, $X$ an $n$-element set and $\mathscr{A}, \mathscr{B}\subset 2^X$ two families. If $|A\cup B|\le s$ for all $A\in\mathscr{A}, B\in \mathscr{B}$, then $\mathscr{A}$ and $\mathscr{B}$ are called cross $s$-union. Assuming that neither $\mathscr{A}$ nor $\mathscr{B}$ is empty, we prove several best possible bounds. In particular, we show that $|\mathscr{A}|+|\mathscr{B}|\le 1+\sum\limits_{0\le i\le s}{{n}\choose{i}}$. Supposing $n\ge 2s$ and $\mathscr{A},\mathscr{B}$ are antichains, we show that $|\mathscr{A}|+|\mathscr{B}|\le {{n}\choose{1}}+{{n}\choose{s-1}}$ unless $\mathscr{A}=\{\emptyset\}$ or $\mathscr{B}=\{\emptyset\}$. An analogous result for three families is established as well.