a b s t r a c tAnderson and Griggs proved independently that a rank-symmetric-unimodal normalized matching (NM) poset possesses a nested chain decomposition (or nesting), and Griggs later conjectured that this result still holds if we remove the condition of rank-symmetry. We give several methods for constructing nestings of rank-unimodal NM posets of rank 3, which together produce substantial progress towards the rank 3 case of the Griggs nesting conjecture. In particular, we show that certain nearly symmetric posets are nested; we show that certain highly asymmetric rank 3 NM posets are nested; and we use results on minimal rank 1 NM posets to show that certain other rank 3 NM posets are nested.
More than twenty-five years ago, Manickam, Miklós, and Singhi conjectured that for positive integers n, k with n ≥ 4k, 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 ≥ 8k 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 ≥ 3k, 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.
Let [n] = {1, 2, . . . , n} and let 2 [n] be the collection of all subsets of [n] ordered by inclusion. C ⊆ 2 [n] is a cutset if it meets every maximal chain in 2 [n] , and the width of C ⊆ 2 [n] is the minimum number of chains in a chain decomposition of C. Fix 0 ≤ m ≤ l ≤ n. What is the smallest value of k such that there exists a cutset that consists only of subsets of sizes between m and l, and such that it contains exactly k subsets of size i for each m ≤ i ≤ l? The answer, which we denote by g n (m, l), gives a lower estimate for the width of a cutset between levels m and l in 2 [n] . After using the Kruskal-Katona Theorem to give a general characterization of cutsets in terms of the number and sizes of their elements, we find lower and upper bounds (as well as some exact values) for g n (m, l).
Let 2 [n] be the poset of all subsets of a set with n elements ordered by inclusion. A long chain in this poset is a chain of n&1 subsets starting with a subset with one element and ending with a subset with n&1 elements. In this paper we prove: Given any collection of at most n&2 skipless chains in 2 [n] , there exists at least one (but sometimes not more than one) long chain disjoint from the chains in the collection. Furthermore, for k 3, given a collection of n&k skipless chains in 2 [n] , there are at least k pairwise disjoint long chains which are also disjoint from the given chains.
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