Shahriar Shahriari

Partitioning the Boolean Lattice into Chains of Large Minimum Size

Let 2n] denote the Boolean lattice of order n, that is, the poset of subsets of {1, ?, n} ordered by inclusion. Recall that 2n] 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 Furedi, we show that there exists a function d(n)~12n such that for any n?0, 2n] may be partitioned into (n?n/2?) chains of size at least d(n). (For comparison, a positive answer to Furedis question would imply that the same result holds for some d(n)~?/2n.) 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 2n] 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.

Long Symmetric Chains in the Boolean Lattice

Let 2n]be the poset of all subsets of a set with n elements ordered by inclusion. A long chain in this poset is a chain ofn?1 subsets starting with a subset with one element and ending with a subset withn?1 elements. In this paper we prove: Given any collection of at mostn?2 skipless chains in 2n], there exists at least one (but sometimes not more than one) long chain disjoint from the chains in the collection. Furthermore, fork?3, given a collection ofn?kskipless chains in 2n], there are at leastkpairwise disjoint long chains which are also disjoint from the given chains.

Partitioning the boolean lattice into a minimal number of chains of relatively uniform size

Let 2[n] denote the Boolean lattice of order n, that is, the poset of subsets of {1,..., n} ordered by inclusion. Extending our previous work on a question of Furedi, we show that for any c > 1, there exist functions e(n) ∼ √n/2 and f(n)∼ c√n log n and an integer N (depending only on c) such that for all n < N, there is a chain decomposition of the Boolean lattice 2[n] into (n ⌊n/2⌋) chains, all of which have size between e(n) and f(n). (A positive answer to Furedis question would imply that the same result holds for some e(n) ∼ √π/2 √n and f(n) = e(n) + 1.) The main tool used is an apparently new observation about rank-collection in normalized matching (LYM) posets.

The Manickam-Miklós-Singhi conjectures for sets and vector spaces

More than twenty-five years ago, Manickam, Miklos, and Singhi conjectured that for positive integers n , k with n ? 4 k , every set of n real numbers with nonnegative sum has at least ( n - 1 k - 1 ) k-element subsets whose sum is also nonnegative. We verify this conjecture when n ? 8 k 2 , which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when k < 10 45 .Moreover, our arguments resolve the vector space analogue of this conjecture. Let V be an n-dimensional vector space over a finite field. Assign a real-valued weight to each 1-dimensional subspace in V so that the sum of all weights is zero. Define the weight of a subspace S ? V to be the sum of the weights of all the 1-dimensional subspaces it contains. We prove that if n ? 3 k , then the number of k-dimensional subspaces in V with nonnegative weight is at least the number of k-dimensional subspaces in V that contain a fixed 1-dimensional subspace. This result verifies a conjecture of Manickam and Singhi from 1988.

Diamond-Free Subsets in the Linear Lattices

Four distinct elements a, b, c, and d of a poset form a diamond if a<b<d

A new matching property for posets and existence of disjoint chains

a< b<d

On the structure of maximum 2-part Sperner families

and a<c<d

Games of Chains and Cutsets in the Boolean Lattice II

a<c<d

On the f-vectors of Cutsets in the Boolean Lattice

. A subset of a poset is diamond-free if no four elements of the subset form a diamond. Even in the Boolean lattices, finding the size of the largest diamond-free subset remains an open problem. In this paper, we consider the linear lattices—poset of subspaces of a finite dimensional vector space over a finite field of order q—and extend the results of Griggs et al. (J. Combin. Theory Ser. A 119(2):310–322, 2012) on the Boolean lattices, to prove that the number of elements of a diamond-free subset of a linear lattice can be no larger than 2+1q+1

Unique factorization in generalized power series rings

2+\frac {1}{q+1}