On Regular Set Systems Containing Regular Subsystems

TL;DR

This paper investigates the conditions under which regular set systems with certain properties can be embedded into larger regular set systems, extending previous algebraic and combinatorial results.

Contribution

It provides new combinatorial techniques to nearly settle the case for h=4 and advances understanding of embedding regular set systems with specific parameters.

Findings

Established necessary and sufficient conditions for embedding regular set systems.

Developed combinatorial methods to address the case h=4.

Connected the problem to partial symmetric Latin squares.

Abstract

Let $X,Y$ be finite sets, $r,s,h, \lambda \in \mathbb{N}$ with $s\geq r, X\subsetneq Y$. By $\lambda \binom{X}{h}$ we mean the collection of all $h$-subsets of $X$ where each subset occurs $\lambda$ times. A coloring of $\lambda\binom{X}{h}$ is {\it $r$-regular} if in every color class each element of $X$ occurs $r$ times. A one-regular color class is a {\it perfect matching}. We are interested in the necessary and sufficient conditions under which an $r$-regular coloring of $\lambda \binom{X}{h}$ can be embedded into an $s$-regular coloring of $\lambda \binom{Y}{h}$. Using algebraic techniques involving glueing together orbits of a suitably chosen cyclic group, the first author and Newman (Combinatorica 38 (2018), no. 6, 1309--1335) solved the case when $\lambda=1,r=s, \gcd (|X|,|Y|,h)=\gcd(|Y|,h)$. Using purely combinatorial techniques, we nearly settle the case $h=4$. Two major challenges include finding all the necessary conditions, and obtaining the exact bound for $|Y|$. It is worth noting that completing partial symmetric latin squares is closely related to the case $\lambda =r=s=1, h=2$ which was solved by Cruse (J. Comb. Theory Ser. A 16 (1974), 18--22).