Forbidding Hamilton cycles in uniform hypergraphs

Abstract

For 1≤d≤ℓ<k, we give a new lower bound for the minimum d-degree threshold that guarantees a Hamilton ℓ-cycle in k-uniform hypergraphs. When k≥4 and d<ℓ=k−1, this bound is larger than the conjectured minimum d-degree threshold for perfect matchings and thus disproves a well-known conjecture of Rödl and Ruciński. Our (simple) construction generalizes a construction of Katona and Kierstead and the space barrier for Hamilton cycles.