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.

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) ' 1 2`n such that for any n \ 0, 2[n] may be partitioned into ( n Nn/2M ) 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) '`p/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 T j (n) with the normalized matching property and log-concave rank numbers. We then use our results on T j (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.

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Let P be a graded poset. Assume that x 1 ; y; x m are elements of rank k and y 1 ; y; y m are elements of rank l for some kol: Further suppose x i py i ; for 1pipm: Lehman and Ron (J. Combin. Theory Ser. A 94 (2001) 399) proved that, if P is the subset lattice, then there exist m disjoint skipless chains in P that begin with the x's and end at the y's. One complication is that it may not be possible to have the chains respect the original matching and hence, in the constructed set of chains, x i and y i may not be in the same chain. In this paper, by introducing a new matching property for posets, called shadow-matching, we show that the same property holds for a much larger class of posets including the divisor lattice, the subspace lattice, the lattice of partitions of a finite set, the intersection poset of a central hyperplane arrangement, the face lattice of a convex polytope, the lattice of noncrossing partitions, and any geometric lattice.