Equality for every n in the suspension Turán bound

Determine whether equality ex(n,Ť) = f(n,k) holds for every n for each balanced tree T whose smaller color class has size k, beyond the cases in which T contains a matching covering one color class.

Background

Theorem 3 establishes, under the Erdős–Sós conjecture for all subtrees of T, that ex(n,Ť) ≤ f(n,k) for sufficiently large n and that equality holds for infinitely many n. Proposition 6 further proves equality for every n when T contains a matching of size k, equivalently a matching covering the smaller color class.

For the remaining balanced trees, the concluding remarks explicitly leave unresolved whether the upper bound is attained for every n. Such a result would require constructing suitable regular graphs in the decomposition-family framework.