Cutsets and anti-chains in linear lattices

Authors:
Bruce Rothschild;Shahriar Shahriari
Affiliations:
Department of Mathematics, University of California, Los Angeles, CA;Department of Mathematics, Pomona College, Claremont, CA
Venue:
Journal of Combinatorial Theory Series A - Special issue in honor of Jacobus H. van Lint
Year:
2006

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Abstract

Consider the poset, ordered by inclusion, of subspaces of a four-dimensional vector space over a field with 2 elements. We prove that, for this poset, any cutset (i.e., a collection of elements that intersects every maximal chain) contains a maximal anti-chain of the poset. In analogy with the same result by Duffus, Sands, and Winkler [D. Duffus, B. Sands, P. Winkler, Maximal chains and anti-chains in Boolean lattices, SIAM J. Discrete Math. 3 (2) (1990) 197-205] for the subset lattice, we conjecture that the above statement holds in any dimension and for any finite base field, and we prove some special cases to support the conjecture.