HAMILTON `-CYCLES IN k-GRAPHS

We say that a k-uniform hypergraph C is an `-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of adjacent edges (in the natural ordering of the edges) intersects in precisely ` vertices. We prove that if 1 ≤ ` < k and k − ` does not divide k then any k-uniform hypergraph on n vertices with minimum degree at least n d k k−` e(k−`) +o(n) contains a Hamilton `-cycle. This confirms a conjecture of Hàn and Schacht. Together with results of Rödl, Ruciński and Szemerédi, our result asymptotically determines the minimum degree which forces an `-cycle for any ` with 1 ≤ ` < k.