Multidimensional Kruskal-Katona theorem
Combinatorics
2011-11-08 v2
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.
Cite
@article{arxiv.1009.2375,
title = {Multidimensional Kruskal-Katona theorem},
author = {Boris Bukh},
journal= {arXiv preprint arXiv:1009.2375},
year = {2011}
}
Comments
8 pages, 1 figure, typos fixed