Short directed cycles in bipartite digraphs

The Caccetta-H\"aggkvist conjecture implies that for every integer $k\ge 1$, if $G$ is a bipartite digraph, with $n$ vertices in each part, and every vertex has out-degree more than $n/(k+1)$, then $G$ has a directed cycle of length at most $2k$. If true this is best possible, and we prove this for $k = 1,2,3,4,6$ and all $k\ge 224,539$. More generally, we conjecture that for every integer $k\ge 1$, and every pair of reals $\alpha, \beta> 0$ with $k\alpha +\beta>1$, if $G$ is a bipartite digraph with bipartition $(A,B)$, where every vertex in $A$ has out-degree at least $\beta|B|$, and every vertex in $B$ has out-degree at least $\alpha|A|$, then $G$ has a directed cycle of length at most $2k$. This implies the Caccetta-H\"aggkvist conjecture (set $\beta>0$ and very small), and again is best possible for infinitely many pairs $(\alpha,\beta)$. We prove this for $k = 1,2$, and prove a weaker statement (that $\alpha+\beta>2/(k+1)$ suffices) for $k=3,4$

Piercing axis-parallel boxes

Let \F be a finite family of axis-parallel boxes in $\R^d$ such that \F contains no $k+1$ pairwise disjoint boxes. We prove that if \F contains a subfamily \M of $k$ pairwise disjoint boxes with the property that for every F\in \F and M\in \M with $F \cap M \neq \emptyset$, either $F$ contains a corner of $M$ or $M$ contains $2^{d-1}$ corners of $F$, then \F can be pierced by $O(k)$ points. One consequence of this result is that if $d=2$ and the ratio between any of the side lengths of any box is bounded by a constant, then \F can be pierced by $O(k)$ points. We further show that if for each two intersecting boxes in \F a corner of one is contained in the other, then \F can be pierced by at most $O(k\log\log(k))$ points, and in the special case where \F contains only cubes this bound improves to $O(k)$

Pure pairs. I. Trees and linear anticomplete pairs

The Erdos-Hajnal Conjecture asserts that for every graph H there is a constant c > 0 such that every graph G that does not contain H as an induced subgraph has a clique or stable set of cardinality at least |G|^c. In this paper, we prove a conjecture of Liebenau and Pilipczuk, that for every forest H there exists c > 0, such that every graph G contains either an induced copy of H, or a vertex of degree at least c|G|, or two disjoint sets of at least c|G| vertices with no edges between them. It follows that for every forest H there is c > 0 so that if G contains neither H nor its complement as an induced subgraph then there is a clique or stable set of cardinality at least |G|^c