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G. Hurlbert Extremal Sets
Extremal Sets

My interest here is mainly on shadows of set systems. For a family $\cal F$ of sets of size $t$, is shadow $\partial \cal F$ consists of all sets of size $t-1$ that are contained in some set of $\cal F$. The Kruskal-Katona theorem says that every family of $m$ sets has a shadow of size at least $k(m)$, where $k(m)$ is explicitly described but cumbersome to compute. I used this theorem in paper with Kostochka and Talysheva on poset dimension in the subset lattice. Lovasz replaced $k(m)$ by a much simpler function $l(m)$ to compute. The Clements-Linstrom and Macauley theorems are analogs of the Kruskal-Katona result for multisets, and I was able to prove a Lovasz-type version and include it in a paper with Bekmetjev, Brightwell, and Czygrinow on thresholds in the multiset lattice (analogous to the Bollobas-Thomason theorem for the subset lattice, which used Lovasz's estimate). This paper (which proved the existence of pebbling thresholds for every graph sequence), is included in the special volume Discrete Mathematics, Editor's Choice, Edition 2003, distinguished by the editors as among the 12 best of the 210 papers published that year. Currently I am working on shadow and threshold results in other lattices, as well as Erdos-Ko-Rado type generalizations involving graphs. My most recent work produced the first injective proof of the famous Erdos-Ko-Rado theorem.