# Generalized Turán problems for $K_{2,t}$

### Abstract

The generalized Turán function $\mathrm{ex}(n,H,F)$ denotes the largest number of copies of $H$ among $F$-free $n$-vertex graphs. We study $\mathrm{ex}(n,H,F)$ when $H$ or $F$ is $K_{2,t}$. We determine the order of magnitude of $\mathrm{ex}(n,H,K_{2,t})$ when $H$ is a tree, and determine its asymptotics for a large class of trees. We also determine the asymptotics of $\mathrm{ex}(n,K_{2,t},F)$ when $F$ has chromatic number at least three and when $F$ is bipartite with one part of order at most two.