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 $t1$ that are contained in some
set of $\cal F$.
The KruskalKatona 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
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 ClementsLinstrom and Macauley theorems are analogs of the
KruskalKatona result for multisets, and I was able to prove
a Lovasztype version and include it in a
with Bekmetjev, Brightwell, and Czygrinow on thresholds in the
multiset lattice (analogous to the BollobasThomason 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
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 ErdosKoRado type
involving graphs. My
produced the first injective proof of the famous ErdosKoRado theorem.
