On the rank of higher inclusion matrices

Abstract

Abstract. Let r ≥ s ≥ 0 be integers and G be an r-graph. The higher inclusion matrix M r s (G) is a {0, 1}-matrix with rows indexed by the edges of G and columns indexed by the subsets of V (G) of size s: the entry corresponding to an edge e and a subset S is 1 if S ⊆ e and 0 otherwise. Following a question of Frankl and Tokushige and a result of Keevash, we define the rank-extremal function rex(n, t, r, s) as the maximum number of edges of an r-graph G having rk M r s (G) ≤ n s − t. For t at most linear in n we determine this function as well as the extremal r-graphs. The special case t = 1 answers a question of Keevash.

Resilience of ranks of higher inclusion matrices

“…More precisely, if a family F of r-subsets of [n] satisfies the condition that |F c | ≤ n−1 r , then rank K (W F r,s ) = rank K (W r,s ) for any field K. Note that a less restrictive bound on |F c | was obtained in [10] when K is a field of characteristic zero. More precisely, if char(K) = 0 and n is large, it was shown in [10] that rank K (W F r,s ) = rank K (W r,s ) for all families F of r-subsets of [n] satisfying that |F c | < n−s r−s . Therefore the following question arises naturally: does Theorem 4 remain true under the assumption that |F c | < n−s r−s ?…”

“…Keevash [17] went further to ask whether Theorem 3 remains true under the assumption that | [n] r \ F | < n−s r−s . This question was answered in the affirmative by Grosu, Person and Szabó [10] for n large (compared with r and s). In the end of [10], the authors remarked that rank resilience property of the higher inclusion matrices has not been studied over fields of positive characteristic.…”

“…This question was answered in the affirmative by Grosu, Person and Szabó [10] for n large (compared with r and s). In the end of [10], the authors remarked that rank resilience property of the higher inclusion matrices has not been studied over fields of positive characteristic. In this paper we prove that the rank of W r,s is resilient over any field K. In fact, the following theorem shows that if the size of F is close to n r then rank K (W F r,s ) = rank K (W r,s ) for an arbitrary field K. To simplify notation, for any family F of r-subsets of [n], we use F c to denote the complement of F in [n] r .…”

“…Let F be a family of r-subspaces of V and K a field with char(K) = p. If |F c | ≤ n r − 1 then rank K (W F r,s (q)) = rank K (W r,s (q)). The techniques we use to prove Theorem 4 and Theorem 6 are completely different from those used by Keevash in [16] and Grosu, Person and Szabó in [10]. The main tool we use to prove Theorem 4 is Bier's bases which give a diagonal form of the higher inclusion matrix W r,s .…”

“…). The techniques we use to prove Theorem 4 and Theorem 6 are completely different from those used by Keevash in [16] and Grosu, Person and Szabó in [10]. The main tool we use to prove Theorem 4 is Bier's bases which give a diagonal form of the higher inclusion matrix W r,s .…”