Ramsey-type results on singletons, co-singletons and monotone sequences in large collections of sets

TL;DR

This paper establishes Ramsey-type theorems for large collections of sets, guaranteeing the existence of specific substructures like singletons, co-singletons, or ordered sets, with exact bounds in some cases.

Contribution

It proves new Ramsey-type results for collections of sets, including exact bounds for the size needed to find particular incidence matrix configurations.

Findings

Existence of a threshold S(n) for certain set configurations

Identification of conditions for finding singletons, co-singletons, or ordered sets

Exact bounds for the number of sets required in some cases

Abstract

We say that a 0-1 matrix $N$ of size $a\times b$ can be found in a collection of sets $\mathcal{H}$ if we can find sets $H_{1}, H_{2}, \dots, H_{a}$ in $\mathcal{H}$ and elements $e_1, e_2, \dots, e_b$ in $\cup_{H \in \mathcal{H}} H$ such that $N$ is the incidence matrix of the sets $H_{1}, H_{2}, \dots, H_{a}$ over the elements $e_1, e_2, \dots, e_b$. We prove the following Ramsey-type result: for every $n\in \N$, there exists a number S(n) such that in any collection of at least S(n) sets, one can find either the incidence matrix of a collection of $n$ singletons, or its complementary matrix, or the incidence matrix of a collection of $n$ sets completely ordered by inclusion. We give several results of the same extremal set theoretical flavour. For some of these, we give the exact value of the number of sets required.