Equality in a result of Kleitman

AbstractAn upset is a set U of subset of a finite set. S such that if U ⊆ V and U ϵ U, then V ϵ U. A downset D is defined analogously. In 1966, Kleitman (J. Combin. Theory 1 (1966), 153–155) proved that if U and D are arbitrary up- and downsets, respectively, then |U| |D| ⩾ 2|S| |U ∩ D|. In this note, we show that a necessary and sufficient condition for equality to hold is: for every minimal element U of U and every maximal element D of D, U ⊆ D. This result is extended to some related inequalities

On the average rank of LYM-sets

Let S be a finite set with some rank function r such that the Whitney numbers wi = |{x S|r(x) = i}| are log-concave. Given so that wk − 1 < wk wk + m, set W = wk + wk + 1 + … + wk + m. Generalizing a theorem of Kleitman and Milner, we prove that every F S with cardinality |F| W has average rank at least kwk + … + (k + m) wk + m/W, provided the normalized profile vector x1, …, xn of F satisfies the following LYM-type inequality: x0 + x1 + … + xn m + 1

On a combinatorial problem of Erdos, Kleitman and Lemke

In this paper, we study a combinatorial problem originating in the following conjecture of Erdos and Lemke: given any sequence of n divisors of n, repetitions being allowed, there exists a subsequence the elements of which are summing to n. This conjecture was proved by Kleitman and Lemke, who then extended the original question to a problem on a zero-sum invariant in the framework of finite Abelian groups. Building among others on earlier works by Alon and Dubiner and by the author, our main theorem gives a new upper bound for this invariant in the general case, and provides its right order of magnitude.Comment: 15 page