EKR sets for large $n$ and $r$

TL;DR

This paper investigates which subsets X of [n] satisfy a generalized Erdős-Ko-Rado property for compressed, intersecting families, providing classification results, bounds on n, and an efficient computational method.

Contribution

It classifies EKR sets X for large n, sharpens bounds for n in terms of r, and introduces a generating function for faster computation of |A(X)|.

Findings

For large n, specifically quadratic in r, the EKR property holds.

When A has a maximal element, the bound on n improves to n > φ²r.

A generating function is provided to compute |A(X)| efficiently.

Abstract

Let $\A\subset\binom{[n]}{r}$ be a compressed, intersecting family and let $X\subset[n]$. Let $\A(X)={A\in\A:A\cap X\ne\emptyset}$ and $\S_{n,r}=\binom{[n]}{r}({1})$. Motivated by the Erd\H{o}s-Ko-Rado theorem, Borg asked for which $X\subset[2,n]$ do we have $|\A(X)|\le|\S_{n,r}(X)|$ for all compressed, intersecting families $\A$? We call $X$ that satisfy this property EKR. Borg classified EKR sets $X$ such that $|X|\ge r$. Barber classified $X$, with $|X|\le r$, such that $X$ is EKR for sufficiently large $n$, and asked how large $n$ must be. We prove $n$ is sufficiently large when $n$ grows quadratically in $r$. In the case where $\A$ has a maximal element, we are able to sharpen this bound to $n>\varphi^{2}r$ implies $|\A(X)|\le|\S_{n,r}(X)|$. We conclude by giving a generating function that speeds up computation of $|\A(X)|$ in comparison with the na\"{i}ve methods.