Saturated chains of subsets and a random walk
Journal of Combinatorial Theory Series A
Discrete Mathematics - First Japan Conference on Graph Theory and Applications
Matchings, cutsets, and chain partitions in graded posets
Discrete Mathematics
Sperner theory
Bipartite graphs and their applications
Bipartite graphs and their applications
Partitioning Boolean lattices into antichains
Discrete Mathematics
Partitioning the boolean lattice into a minimal number of chains of relatively uniform size
European Journal of Combinatorics
A strategy for efficient crawling of rich internet applications
ICWE'11 Proceedings of the 11th international conference on Web engineering
| Hi-index | 0.00 |
Let 2[n] denote the Boolean lattice of order n, that is, the poset of subsets of {1, ..., n} ordered by inclusion. Recall that 2[n] may be partitioned into what we call the canonical symmetric chain decomposition (due to de Bruijn, Tengbergen, and Kruyswijk), or CSCD. Motivated by a question of Füredi, we show that there exists a function d(n) ∼ ½√n such that for any n ≥ 0, 2[n] may be partitioned into (n/[n/2]) chains of size at least d(n). (For comparison, a positive answer to Füredi's question would imply that the same result holds for some d(n) ∼ √π/2 √n) More precisely, we first show that for 0 ≤ j ≤ n, the union of the lowest j + 1 elements from each of the chains in the CSCD of 2[n] forms a poset Tj(n) with the normalized matching property and log-concave rank numbers. We then use our results on Tj(n) to show that the nodes in the CSCD chains of size less than 2d(n) may be repartitioned into chains of large minimum size, as desired.