An Exact Result for Hypergraphs and Upper Bounds for the Turán Density of K

We first answer a question of de Caen [Extremal Problems for Finite Sets, János Bolyai Math. Soc., Budapest, 1994, pp. 187–197]: given r ≥ 3, if G is an r-uniform hypergraph on n vertices such that every r + 1 vertices span 1 or r + 1 edges, then G = Kr n or K r n−1, assuming that n > (p − 1)r, where p is the smallest prime factor of r − 1. We then show that the Turán density π(Kr r+1) ≤ 1 − 1/r − (1 − 1/rp−1)(r − 1)2/(2rp( ( r+p p−1 ) + ( r+1 2 ) )), for all even r ≥ 4, improving a well-known bound 1 − 1 r of de Caen [Ars Combin., 16 (1983), pp. 5–10] and Sidorenko [Vestnik Moskov. Univ. Ser. I Mat. Mekh., 76 (1982), pp. 3–6].