A well-known result of Ajtai et al. from 1982 states that every $k$-graph $H$ on $n$ vertices, with girth at least five, and average degree $t^{k-1}$ contains an independent set of size $c n \frac{(\log t)^{1/(k-1)}}{t}$ for some $c>0$. In this talk, we explore a related problem where we relax the girth condition, allowing certain cycles of length 2, 3, and 4. We will also present lower bounds on the size of independent sets in hypergraphs under specific degree conditions. This is joint work with Vojtěch Rödl and Yi Zhao.