Turán upper bound for suspensions of arbitrary trees

Prove that for every tree T whose smaller bipartition class has size k, the inequality ex(n, ̂T) ≤ f(n,k) holds for all sufficiently large n.

Background

The main theorem is proved for balanced trees, meaning that the two color classes differ in size by at most one. The authors observe that parts of the argument extend to certain unbalanced trees when the structural decomposition lemma remains valid, and they consequently formulate a conjecture covering every tree. The conjecture concerns the same function f(n,k) used in the main theorem, with k equal to the size of the smaller color class of T.

References

Based on this, we conjecture that our result holds for any tree. Conjecture 2. Let T be a tree such that the smaller color class has size k. Then for large n, ex(n, ̂T) ≤ f(n, k).

— Turán problems for suspension of a balanced tree  (2503.05166 - Zhu et al., 7 Mar 2025) in Section 5, Concluding Remarks, Conjecture 2 (page 16)