Hypergraphs with no tight cycles

We show that every r-uniform hypergraph on n vertices which does not contain a tight cycle has has at most O(n^{r-1}(log n)^{5}) edges. This is an improvement on the previously best-known bound, of n^{r-1}e^{O(\sqrt{log n})} due to Sudakov and Tomon, and our proof builds on their work. A recent construction of B. Janzer implies that our bound is tight up to an O((log n)^{4} log log n) factor

Towards a hypergraph version of the P\'osa-Seymour conjecture

We prove that for fixed $r\ge k\ge 2$, every $k$-uniform hypergraph on $n$ vertices having minimum codegree at least $(1-(\binom{r-1}{k-1}+\binom{r-2}{k-2})^{-1})n+o(n)$ contains the $(r-k+1)$th power of a tight Hamilton cycle. This result may be seen as a step towards a hypergraph version of the P\'osa--Seymour conjecture. Moreover, we prove that the same bound on the codegree suffices for finding a copy of every spanning hypergraph of tree-width less than $r$ which admits a tree decomposition where every vertex is in a bounded number of bags.Comment: 22 page

Towards Lehel's conjecture for 4-uniform tight cycles

A $k$-uniform tight cycle is a $k$-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size $k$ formed by $k$ consecutive vertices in the ordering. We prove that every red-blue edge-coloured $K_n^{(4)}$ contains a red and a blue tight cycle that are vertex-disjoint and together cover $n-o(n)$ vertices. Moreover, we prove that every red-blue edge-coloured $K_n^{(5)}$ contains four monochromatic tight cycles that are vertex-disjoint and together cover $n-o(n)$ vertices.Comment: arXiv admin note: text overlap with arXiv:1606.05616 by other author