On $\gamma$-vectors satisfying the Kruskal-Katona inequalities

TL;DR

This paper explores the properties of $b3$-vectors of flag homology spheres, demonstrating they satisfy Kruskal-Katona inequalities in various cases and proposing a conjecture extending these results.

Contribution

It provides explicit examples of complexes with $b3$-vectors satisfying Kruskal-Katona inequalities and formulates a conjecture generalizing Gal's nonnegativity conjecture.

Findings

Examples of complexes with $b3$-vectors satisfying Kruskal-Katona inequalities.

Flag $(d-1)$-spheres with at most $2d+2$ vertices have $b3$-vectors satisfying these inequalities.

Conjecture that all flag homology spheres' $b3$-vectors satisfy Kruskal-Katona inequalities.

Abstract

We present examples of flag homology spheres whose $\gamma$-vectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit simplicial complexes whose $f$-vectors are the $\gamma$-vectors in question. In another direction, we show that if a flag $(d-1)$-sphere has at most $2d+2$ vertices its $\gamma$-vector satisfies the Kruskal-Katona inequalities. We conjecture that if $\Delta$ is a flag homology sphere then $\gamma(\Delta)$ satisfies the Kruskal-Katona inequalities. This conjecture is a significant refinement of Gal's conjecture, which asserts that such $\gamma$-vectors are nonnegative.