Two new results on maximal left-compressed intersecting families

TL;DR

This paper investigates the growth and significance of maximal left-compressed intersecting families, identifying key structures called canonical MLCIFs and extending classical combinatorial theorems to weighted contexts.

Contribution

It proves the doubly-exponential growth of MLCIFs and characterizes canonical MLCIFs as those with maximum weight under certain functions, extending Erdős–Ko–Rado theorem.

Findings

Number of k-uniform MLCIFs grows doubly-exponentially with k

Canonical MLCIFs are the most important and uniquely maximum weight under some functions

Extension of Erdős–Ko–Rado theorem to weighted set systems

Abstract

This paper presents two new results on the theory of maximal left-compressed intersecting families (MLCIFs). First, we answer a question raised by Barber by showing that the number of $k$-uniform MLCIFs on a ground set of size $n$ grows as a doubly-exponential function of $k$, which we identify up to a log factor in the exponent. Among these MLCIFs we identify $k$ specific MLCIFs -- which we call the canonical MLCIFs -- as being in a meaningful way the most important MLCIFs. Specifically, our second main result shows that the canonical MLCIFs are precisely those which can have maximum weight among all $k$-uniform MLCIFs under a non-trivial increasing weight function, and moreover that each canonical MLCIF is the unique $k$-uniform MLCIF of maximum weight for some increasing weight function. This gives an interesting generalisation of the Erd\H{o}s--Ko--Rado theorem to a notion of size which places greater significance on some elements of the ground set than others.