Independence number of hypergraphs under degree conditions

A well-known result of Ajtai Komlós, Pintz, Spencer, and Szemerédi (J. Combin. Theory Ser. 32 (1982), 321–335) states that every k $$ -graph H on n vertices, with girth at least five, average degree t − 1 {t}^{k-1} contains an independent set size c ( log ) / cn{\left(\log t\right)}^{1/\left(k-1\right)}/t for some > 0 c>0 . In this paper we show the same can be found under weaker conditions allowing certain cycles length 2, 3, 4. Our work is motivated by a problem Lo Zhao, who asked ≥ 4 k\ge , how large vertices necessarily has when its maximum 2 \left(k-2\right) -degree Δ ≤ d {\Delta}_{k-2}(H)\le dn (The corresponding respect to \left(k-1\right) -degrees was solved Kostochka, Mubayi, Verstraëte (Random Struct. & Algorithms 44 (2014), 224–239).) c{\left(\frac{n}{d}\mathrm{loglog}\frac{n}{d}\right)}^{1/\left(k-1\right)} additional conditions, c{\left(\frac{n}{d}\log \frac{n}{d}\right)}^{1/\left(k-1\right)} The former assertion gives new upper bound Turán density complete -graphs.