Idealness of k-wise intersecting families
Abstract
A clutter isk-wise intersectingif everykmembers have a common element, yet no element belongs to all members. We conjecture that, for some integer$$k\ge 4$$k≥4, everyk-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for$$k=4$$k=4for 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 As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975.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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Nov 11, 2020
- Source ID
- 10.1007/s10107-020-01587-x
Entities
People
- Ahmad Abdi
- Dabeen Lee
- Gérard Cornuéjols
- Tony Huynh
Organizations
- Australian Research Council
- Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada
- European Research Council
- Institute for Basic Science
- Office of Naval Research