Independence number of hypergraphs under degree conditions

A well‐known result of Ajtai Komlós, Pintz, Spencer, and Szemerédi (J. Combin. Theory Ser. A 32 (1982), 321–335) states that every k$$ k $$ ‐graph H$$ H $$ on n$$ n $$ vertices, with girth at least five, and average degree tk−1$$ {t}^{k-1} $$ contains an independent set of size cn(logt)1/(k−1)/t$$ cn{\left(\log t\right)}^{1/\left(k-1\right)}/t $$ for some c>0$$ c>0 $$ . 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$$ k\ge 4 $$ , how large of an independent set a k$$ k $$ ‐graph H$$ H $$ on n$$ n $$ vertices necessarily has when its maximum (k−2)$$ \left(k-2\right) $$ ‐degree Δk−2(H)≤dn$$ {\Delta}_{k-2}(H)\le dn $$ . (The corresponding problem with respect to (k−1)$$ \left(k-1\right) $$ ‐degrees was solved by Kostochka, Mubayi, and Verstraëte (Random Struct. & Algorithms 44 (2014), 224–239).) In this paper we show that every k$$ k $$ ‐graph H$$ H $$ on n$$ n $$ vertices with Δk−2(H)≤dn$$ {\Delta}_{k-2}(H)\le dn $$ contains an independent set of size cndloglognd1/(k−1)$$ c{\left(\frac{n}{d}\mathrm{loglog}\frac{n}{d}\right)}^{1/\left(k-1\right)} $$ , and under additional conditions, an independent set of size cndlognd1/(k−1)$$ c{\left(\frac{n}{d}\log \frac{n}{d}\right)}^{1/\left(k-1\right)} $$ . The former assertion gives a new upper bound for the (k−2)$$ \left(k-2\right) $$ ‐degree Turán density of complete k$$ k $$ ‐graphs.