Christian Bey, Konrad Engel, Gyula O.H. Katona, Uwe Leck
On the average size of sets in intersecting Sperner families
Preprint series: Preprints aus dem Fachbereich Mathematik, Universitt Rostock

MSC:

05D05 Extremal set theory

06A07 Combinatorics of partially ordered sets

Abstract: We show that the average size of subsets of $[n]$ forming
an intersecting Sperner family of cardinality not less
than $\binom{n-1}{k-1}$ is at least $k$ provided that
$k\le\frac{n}{2}-\frac{\sqrt{n}}{2}+1$. The statement is
not true if
$k> \frac{n}{2}-\frac{\sqrt{8n+1}}{8}+\frac{9}{8}$.
Keywords: intersecting families, Sperner families, Erd\H os-Ko-Rado theorem