Remaining parameter ranges for {P_r, K_{t,t}} and {P_r, K_{2,t}} cases

Ascertain the boundedness of the tree-independence number for the following unresolved parameter ranges: (i) r ≥ 7 and t ≥ 2 in both the {P_r, K_{t,t}}-free and {P_r, K_{2,t}}-free settings, and (ii) r ≥ 5 and t ≥ 3 in the {P_r, K_{t,t}}-free setting.

Background

The paper proves bounded tree-independence for {P_r, K_{2,t}}-free graphs when r ≤ 6, and notes earlier progress for smaller r in the {P_r, K_{t,t}} case. Larger r and specific ranges remain unresolved.

References

Our work leaves open the case $r\ge 7$ and $t \ge 2$ of \zcref{conj:PrKtt,conj:PrK2t}, as well as the case $r\ge 5$ and $t \ge 3$ of \zcref{conj:PrKtt}.

— Tree-independence number and forbidden induced subgraphs: excluding a $6$-vertex path and a $(2,t)$-biclique  (2604.01999 - Chudnovsky et al., 2 Apr 2026) in Section 1 (Introduction)