Infinitely many size-Ramsey numbers of $k$-uniform relaxed $\ell$-trees are not polynomial

Abstract: The size-Ramsey number $\widehat{R}<em>k(\mathcal G)$ of a $k$-uniform hypergraph $\mathcal G$ is the minimum number of edges in a $k$-uniform hypergraph $\mathcal H$ such that every $2$-edge-coloring of $\mathcal H$ contains a monochromatic copy of $\mathcal G$. The following question was pointed out by Fox and recorded by Dudek, La Fleur, Mubayi and Rödl~\cite{Dudek-Fleur-Mubayi-Rodl}: for fixed $2\le \ell&lt;k$, is the size-Ramsey number of every $k$-uniform relaxed $\ell$-tree bounded by a polynomial in $n$? We answer this question in the range [ \ell\geq3 \quad\text{and}\quad \ell+1\leq k\leq2\ell-2. ] For every sufficiently large $n$, we construct a $k$-uniform relaxed $\ell$-tree $\bar{\mathcal{T}}</em>{n,\ell}<sup>{(k)}$ on exactly $n$ vertices such that $$\widehat{R}<em>k(\bar{\mathcal{T}}</em>{n,\ell}^{(k)})\ge 2^{{c_{k,\ell}n^{1/\ell}}}$$ for a constant $c_{k,\ell}&gt;0$ depending only on $k$ and $\ell$.