Frankl-Füredi Type Inequalities for Polynomial Semi-lattices

Abstract

Let $X$ be an $n$-set and $L$ a set of nonnegative integers. ${\cal F}$, a set of subsets of $X$, is said to be an $L$ -intersection family if and only if for all $E \neq F \in {\cal F}, \, |E \cap F | \in L$. A special case of a conjecture of Frankl and Füredi states that if $ L = \{1, 2, \dots,k\}$,$ k$ a positive integer, then $|{\cal F}| \leq\sum_{i=0}^{k}{n-1\choose i}$.

Here $|{\cal F}|$ denotes the number of elements in ${\cal F}$.

Recently Ramanan proved this conjecture. We extend his method to polynomial semi-lattices and we also study some special $L$-intersection families on polynomial semi-lattices.

Finally we prove two modular versions of Ray-Chaudhuri-Wilson inequality for polynomial semi-lattices.