We study the generalized Turán function ex(n,H,F), when H or F is a double star Sa,b, which is a tree with a central edge uv, a leaves connected to u and b leaves connected to v. We determine ex(n,Kk,Sa,b) and ex(n,Sa,b,F) for sufficiently large n, where F is either a 3-chromatic graph with an edge whose deletion results in a bipartite graph, or the 2-fan, i.e. two triangles sharing a vertex. We also give bounds on ex(n,Sa,b,Sc,d).
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Dániel Gerbner (2023) studied this question.