Forbidding Hamilton cycles in uniform hypergraphs

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