Flattening rank and its combinatorial applications

Abstract

Given a d-dimensional tensor T:A1×…×Ad→F (where F is a field), the i-flattening rank of T is the rank of the matrix whose rows are indexed by Ai, columns are indexed by Bi=A1×…×Ai−1×Ai+1×…×Ad and whose entries are given by the corresponding values of T. The max-flattening rank of T is defined as mfrank(T)=maxi∈[d]⁡franki(T). A tensor T:Ad→F is called semi-diagonal, if T(a,…,a)≠0 for every a∈A, and T(a1,…,ad)=0 for every a1,…,ad∈A that are all distinct. In this paper we prove that if T:Ad→F is semi-diagonal, then mfrank(T)≥|A|d−1, and this bound is the best possible.We give several applications of this result, including a generalization of the celebrated Frankl-Wilson theorem on forbidden intersections. Also, addressing a conjecture of Aharoni and Berger, we show that if the edges of an r-uniform multi-hypergraph H are colored with z colors such that each color class is a matching of size t, then H contains a rainbow matching of size t provided z>(t−1)(rtr). This improves previous results of Alon and Glebov, Sudakov, and Szabó.