论文信息 - On the f-vectors of Cutsets in the Boolean Lattice

On the f-vectors of Cutsets in the Boolean Lattice

A cutset in the poset 2[n], of subsets of {1, ?, n} ordered by inclusion, is a subset of 2[n] that intersects every maximal chain. Let 0???1 be a real number. Is it possible to find a cutset in 2[n] that, for each 0?i?n, contains at most ? (ni) subsets of size i? Let ?(n) be the greatest lower bound of all real numbers for which the answer is positive. In this note we prove the rather surprising fact that limn?∞?(n)=0.

Matthew Haines | Shahriar Shahriari

[1] Daniel J. Kleitman,et al. Minimum cutsets for an element of a boolean lattice , 1989 .

[2] Shahriar Shahriari,et al. Long Symmetric Chains in the Boolean Lattice , 1996, J. Comb. Theory, Ser. A.

[3] Charles J. Colbourn,et al. The Combinatorics of Network Reliability , 1987 .

[4] Zoltán Füredi,et al. A minimal cutset of the boolean lattice with almost all members , 1989, Graphs Comb..

[5] Feng Shi,et al. The density of a maximum minimal cut in the subset lattice of a finite set is almost one , 1993, Discret. Math..

[6] Shahriar Shahriari,et al. On Uniform f-vectors of Cutsets in the Truncated Boolean Lattice , 2000, Comb..

[7] Peter Winkler,et al. Maximal Chains and Antichains in Boolean Lattices , 1990, SIAM J. Discret. Math..

[8] Hunter S. Snevily. Combinatorics of finite sets , 1991 .

[9] Shahriar Shahriari,et al. Games of Chains and Cutsets in the Boolean Lattice , 1997 .

[10] Shahriar Shahriari,et al. Games of Chains and Cutsets in the Boolean Lattice II , 2001, Order.

[11] Jerrold R. Griggs. Matchings, cutsets, and chain partitions in graded posets , 1995, Discret. Math..

[12] Richard J. Nowakowski. Cutsets of Boolean lattices , 1987, Discret. Math..

[13] D. Kleitman,et al. Proof techniques in the theory of finite sets , 1978 .