Graphs with the Erdos-Ko-Rado property

For a graph G, vertex v of G and integer r >= 1, we denote the family of independent r-sets of V(G) by I^(r)(G) and the subfamily {A in I^(r)(G): v in A} by I^(r)_v(G); such a family is called a star. Then, G is said to be r-EKR if no intersecting subfamily of I^(r)(G) is larger than the largest star in I^(r)(G). If every intersecting subfamily of I^(r)_v(G) of maximum size is a star, then G is said to be strictly r-EKR. We show that if a graph is r-EKR then its lexicographic product with any complete graph is r-EKR. For any graph G, we define mu(G) to be the minimum size of a maximal independent vertex set. We conjecture that, if 1 <= r <= 1/2 mu(G), then G is r-EKR, and if r < 1/2 mu(G), then G is strictly r-EKR. This is known to be true when G is an empty graph, a cycle, a path or the disjoint union of complete graphs. We show that it is also true when G is the disjoint union of a pair of complete multipartite graphs

Supersaturation and stability for forbidden subposet problems

We address a supersaturation problem in the context of forbidden subposets. A family $\mathcal{F}$ of sets is said to contain the poset $P$ if there is an injection $i:P \rightarrow \mathcal{F}$ such that $p \le_P q$ implies $i(p) \subset i (q)$. The poset on four elements $a,b,c,d$ with $a,b \le c,d$ is called butterfly. The maximum size of a family $\mathcal{F} \subseteq 2^{[n]}$ that does not contain a butterfly is $\Sigma(n,2)=\binom{n}{\lfloor n/2 \rfloor}+\binom{n}{\lfloor n/2 \rfloor+1}$ as proved by De Bonis, Katona, and Swanepoel. We prove that if $\mathcal{F} \subseteq 2^{[n]}$ contains $\Sigma(n,2)+E$ sets, then it has to contain at least $(1-o(1))E(\lceil n/2 \rceil +1)\binom{\lceil n/2\rceil}{2}$ copies of the butterfly provided $E\le 2^{n^{1-\varepsilon}}$ for some positive $\varepsilon$. We show by a construction that this is asymptotically tight and for small values of $E$ we show that the minimum number of butterflies contained in $\mathcal{F}$ is exactly $E(\lceil n/2 \rceil +1)\binom{\lceil n/2\rceil}{2}$

Compression and Erdos-Ko-Rado graphs

For a graph G and integer r >= 1 we denote the collection of independent r-setsof G by I^(r)(G). If v is in V(G) then I^(r)_v(G) is the collection of all independent r-sets containing v. A graph G is said to be r-EKR, for r >= 1, iff no intersecting family A of I^(r)(G) is larger than max_{v in V(G)} |I^(r)_v(G)|. There are various graphs that are known to have this property; the empty graph of order n >= 2r (this is the celebrated Erdos-Ko-Rado theorem), any disjoint union of atleast r copies of K_t for t >= 2, and any cycle. In this paper we show how these results can be extended to other classes of graphs via a compression proof technique. In particular we extend a theorem of Berge, showing that any disjoint union of at least r complete graphs, each of order at least two, is r-EKR. We also show that paths are r-EKR for all r >= 1