Extend bipartite Turán results to further tree families

Determine \(\operatorname{ex}_{b}(a,b,F)\) and \(\operatorname{ex}_{b,c}(a,b,F)\) for further families of trees, in particular for spiders of the form \(S_{k,d^{\ell+1}}\), thereby extending the cases established in Theorems 8 and 9.

Background

The paper obtains exact or sharp results for several infinite classes of trees, including selected spiders, double stars, and certain caterpillars. The authors identify the determination of the bipartite and connected bipartite Turán numbers for additional tree families as an open direction, singling out spiders of the indicated form as a particularly natural extension of their existing theorems.