Tight Cycles in Hypergraphs

We apply a recent version of the Strong Hypergraph Regularity Lemma (see [Allen, P., J. Böttcher, O. Cooley, and R. Mycroft, Regular slices for hypergraphs, Extended abstract submitted to Eurocomb 2015], [Allen, P., J. Böttcher, O. Cooley, and R. Mycroft, Tight cycles and regular slices in dense hypergraphs, Submitted. http://arxiv.org/pdf/1411.4957.]) to prove two new results on tight cycles in k-uniform hypergraphs.The first result is an extension of the Erdős-Gallai Theorem for graphs: For every δ>0, every sufficiently large k-uniform hypergraph on n vertices with at least edges contains a tight cycle of length αn for any α∈[0,1].Our second result concerns k-partite k-uniform hypergraphs with partition classes of size n and for each α∈(0,1) provides an asymptotically optimal minimum codegree requirement for the hypergraph to contain a cycle of length αkn.