Independence number of hypergraphs under degree conditions

A well-known result of Ajtai Komlos, Pintz, Spencer, and Szemeredi (J. Combin. Theory Ser. A 32 (1982), 321-335) states that every k- graph H on n vertices, with girth at least five, and average degree tk- 1 contains an independent set of size cn( log t)1/(k-1)/t for some c > 0. In this paper we show that an independent set of the same size can be found under weaker conditions allowing certain cycles of length 2, 3, and 4. Our work is motivated by a problem of Lo and Zhao, who asked for k = 4, how large of an independent set a k- graph H on n vertices necessarily has when its maximum (k - 2)- degree.k- 2(H) =.. n. (The corresponding problem with respect to (k-1)- degreeswas solved by Kostochka, Mubayi, and Verstraete (Random Struct. & Algorithms 44 (2014), 224-239).) In this paper we show that every k- graph H on n vertices with.k- 2(H) =..n contains an independent set of size c( n.. log log n..) 1/(k-1), and under additional conditions, an independent set of size c( n.. log n..) 1/(k-1). The former assertion gives a new upper bound for the (k - 2)- degree Turan density of complete k- graphs.