Multidimensional Kruskal-Katona theorem

TL;DR

This paper generalizes the Kruskal-Katona theorem to multidimensional settings, providing bounds on the size of shadows of families of tuples in combinatorial structures, extending classical combinatorial results.

Contribution

It introduces a multidimensional version of the Kruskal-Katona theorem, extending Lovasz's work to higher-dimensional tuple families.

Findings

Shadow size of family F is at least binom{x}{r-1}^d when |F|=binom{x}{r}^d.

Generalization applies to families in binom{X}{r}^d with real x≥r.

Provides a lower bound on shadow sizes in multidimensional combinatorial structures.

Abstract

We present a generalization of a version of the Kruskal-Katona theorem due to Lovasz. A shadow of a d-tuple (S_1,...,S_d) in binom{X}{r}^d consists of d-tuples (S_1',...,S_d') in binom{X}{r-1}^d obtained by removing one element from each of S_i. We show that if a family F in binom{X}{r}^d has size |F|=binom{x}{r}^d for a real number x>=r, then the shadow of F has size at least binom{x}{r-1}^d.