Polynomial size-Ramsey numbers for large-uniformity relaxed trees

Prove that for every fixed integer \(\ell\ge 3\) and every integer \(k\ge 2\ell-1\), every \(k\)-uniform relaxed \(\ell\)-tree on \(n\) vertices has size-Ramsey number bounded by \(O(n^{C_{k,\ell}})\) for some constant \(C_{k,\ell}>0\).

Background

The paper proves that, for $\ell\ge 3$ and $\ell+1\le k\le 2\ell-2$, there exist $k$-uniform relaxed $\ell$-trees whose size-Ramsey numbers grow exponentially in $n^{1/\ell}$, and therefore are not polynomial in $n$. The lower-bound construction relies on lifting a graph coloring to a $k$-uniform hypergraph coloring while ensuring that no $k$-edge contains both a red and a blue copy of $K_\ell$.

The method fails when $k\ge 2\ell-1$, because two $\ell$-subsets of a $k$-set may intersect in at most one vertex and therefore may be edge-disjoint as graph cliques. The conjecture asks whether this failure reflects a genuine change in behavior: namely, whether all relaxed $\ell$-trees in this larger-$k$ regime have polynomial size-Ramsey numbers, as strict $\ell$-trees already do.