Idealness of k-wise Intersecting Families

Abstract

A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that every 4-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it in the binary case. Two key ingredients for our proof are Jaeger's 8-flow theorem for graphs, and Seymour's characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. arXiv:1912

Idealness of k-wise intersecting families

“…For x, y ∈ {0, 1} n , x y denotes the coordinate-wise sum of x, y modulo 2. We say that S is a vector space over G F (2), or simply a binary space, if a b ∈ S for all a, b ∈ S. Notice that a nonempty binary space necessarily contains 0.…”

“…Let E be a finite set, S ⊆ {0, 1} E a binary space, and S ⊥ the orthogonal complement of S, that is, S ⊥ = {y ∈ {0, 1} E : y x ≡ 0 (mod 2) ∀x ∈ S}. Notice that S ⊥ is another binary space, and that (S ⊥ ) ⊥ = S. Therefore, there exists a 0-1 matrix A whose columns are labeled by E such that S = x ∈ {0, 1} E : Ax ≡ 0 (mod 2) , and S ⊥ is the row space of A generated over G F (2).…”

“…Then M(V 8 ) has a 3-cycle cover given by the following cuts of V 8 : δ({1, 6, 7, 8}), δ({1, 7}), δ({2, 4}), where δ(X ) is the set of edges with exactly one end in X . i ∈ [2]. Similarly, we may assume e / ∈ C i 3 for i ∈ [2].…”

“…i ∈ [2]. Similarly, we may assume e / ∈ C i 3 for i ∈ [2]. But now {C 1 j C 2 j : j ∈ [3]} is a 3-cycle cover of M. For (3), suppose E 1 ∩ E 2 = {e, f , g}.…”

“…Let A be the × (2 − 1) matrix whose columns are all the nonzero vectors in {0, 1} . The binary matroid represented by A is called a projective geometry over G F (2), and is denoted PG( − 1, 2). See Fig.…”