Intersecting faces of a simplicial complex via algebraic shifting

TL;DR

This paper proves Borg's conjecture on the size of $t$-intersecting faces in simplicial complexes using algebraic shifting, extending results to sequentially Cohen-Macaulay $i$-near-cones.

Contribution

It provides a new proof of Borg's conjecture for shifted complexes and verifies it for sequentially Cohen-Macaulay $i$-near-cones using algebraic shifting.

Findings

Borg's conjecture holds for shifted complexes.

Verification of Borg's conjecture for sequentially Cohen-Macaulay $i$-near-cones.

New proof techniques based on algebraic shifting.

Abstract

A family $\mathcal{A}$ of sets is {\it $t$-intersecting} if the cardinality of the intersection of every pair of sets in $\mathcal{A}$ is at least $t$, and is an {\it $r$-family} if every set in $\mathcal{A}$ has cardinality $r$. A well-known theorem of Erd\H{o}s, Ko, and Rado bounds the cardinality of a $t$-intersecting $r$-family of subsets of an $n$-element set, or equivalently of $(r-1)$-dimensional faces of a simplex with $n$ vertices. As a generalization of the Erd\H{o}s-Ko-Rado theorem, Borg presented a conjecture concerning the size of a $t$-intersecting $r$-family of faces of an arbitrary simplicial complex. He proved his conjecture for shifted complexes. In this paper we give a new proof for this result based on work of Woodroofe. Using algebraic shifting we verify Borg's conjecture in the case of sequentially Cohen-Macaulay $i$-near-cones for $t=i$.