Two new results on maximal left-compressed intersecting families

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.