Sunflowers in Lattices

Abstract

A Sunflower is a subset $S$ of a lattice, with the property that the meet of any two elements in $S$ coincides with the meet of all of $S$. The Sunflower Lemma of Erdös and Rado asserts that a set of size at least $1 + k!(t-1)^k$ of elements of rank $k$ in a Boolean Lattice contains a sunflower of size $t$. We develop counterparts of the Sunflower Lemma for distributive lattices, graphic matroids, and matroids representable over a fixed finite field. We also show that there is no counterpart for arbitrary matroids.