Determine the bipartite Turán numbers of all seven-vertex trees

Determine \(\operatorname{ex}_{b}(a,b,F)\) and \(\operatorname{ex}_{b,c}(a,b,F)\) for every tree \(F\) on seven vertices, with the aim of identifying proofs that extend to infinite families of trees.

Background

The paper completely determines the relevant bipartite and connected bipartite Turán numbers for trees up to six vertices, apart from the stated feasibility conventions and small exceptional cases. The next natural finite classification problem is therefore to determine both functions for all trees on seven vertices, potentially revealing arguments applicable to broader infinite classes.

References

Problem 15. Determine exb (a, b, F) and exb,c(a, b, F) for all the trees on 7 vertices, with the hope that some of the proofs extend to infinite families of trees (as was the case that led to Theorems 8, 9, and 10).

— Bipartite Turán number of trees  (2502.09052 - Caro et al., 13 Feb 2025) in Problem 15, Section 4, “Concluding Remarks and Open Problems”