Idealness of k-wise intersecting families

• 成果类型: Article
• 署名作者: Abdi, Ahmad; Cornuejols, Gerard; Huynh, Tony; Lee, Dabeen
• 署名单位: University of London; London School Economics & Political Science; Carnegie Mellon University; Monash University; Institute for Basic Science - Korea (IBS)
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-020-01587-x
• 发表日期: 2022
• 页码: 29-50
• 关键词: graphs; cycle

摘要

A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that, for some integer k >= 4, every k-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for k = 4 for the class of binary clutters. 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 As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. We also discuss connections to the chromatic number of a clutter, projective geometries over the two-element field, uniform cycle covers in graphs, and quarter- integral packings of value two in ideal clutters.