Union-intersecting set systems

Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of any t sets. The maximal size of such a set system is determined exactly if s+t4. Finally, we exactly determine the maximal size of a k-uniform set system that has the above described (s,t)-union-intersecting property, for large enough n.Comment: 9 page

A note on vertex Tur\'an problems in the Kneser cube

The Kneser cube $Kn_n$ has vertex set $2^{[n]}$ and two vertices $F,F'$ are joined by an edge if and only if $F\cap F'=\emptyset$. For a fixed graph $G$, we are interested in the most number $vex(n,G)$ of vertices of $Kn_n$ that span a $G$-free subgraph in $Kn_n$. We show that the asymptotics of $vex(n,G)$ is $(1+o(1))2^{n-1}$ for bipartite $G$ and $(1-o(1))2^n$ for graphs with chromatic number at least 3. We also obtain results on the order of magnitude of $2^{n-1}-vex(n,G)$ and $2^n-vex(n,G)$ in these two cases. In the case of bipartite $G$, we relate this problem to instances of the forbidden subposet problem

Eötvös Collegium – Collegiumi Értesítő (2013/2014)

Az Eötvös Collegium évkönyvének IV. száma. A kiadvány a "Az Oktatási Hivatal által nyilvántartott szakkolégiumok támogatása" című pályázat keretében (NTP-SZKOLL-13-0030) valósult meg