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Idealness of k-wise intersecting families

Abdi, Ahmad ORCID: 0000-0002-3008-4167, Cornuejols, Gerard, Huynh, Tony and Lee, Dabeen (2022) Idealness of k-wise intersecting families. Mathematical Programming, 192 (1-2). 29 - 50. ISSN 0025-5610

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Identification Number: 10.1007/s10107-020-01587-x


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 As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975.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.

Item Type: Article
Official URL:
Additional Information: © 2020 The Authors
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 27 Oct 2020 10:54
Last Modified: 12 Jul 2024 06:06

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