A lower bound on the multicolor size-Ramsey numbers of paths in hypergraphs
The $r$-color size-Ramsey number of a $k$-uniform hypergraph $H$, denoted by $\hat{R}_r(H)$, is the minimum number of edges in a $k$-uniform hypergraph $G$ such that for every $r$-coloring of the edges of $G$ there exists a monochromatic copy of $H$. In the case of $2$-uniform paths $P_n$, it is known that $Ω(r^2n)=\hat{R}_r(P_n)=O((r^2\log r)n)$ with the best bounds essentially due to Krivelevich. In a recent breakthrough result, Letzter, Pokrovskiy, and Yepremyan gave a linear upper bound on the $r$-color size-Ramsey number of the $k$-uniform tight path $P_{n}^{(k)}$; i.e. $\hat{R}_r(P_{n}^{(k)})=O_{r,k}(n)$. Winter gave the first non-trivial lower bounds on the 2-color size-Ramsey number of $P_{n}^{(k)}$ for $k\geq 3$; i.e. $\hat{R}_2(P_{n}^{(3)})\geq \frac{8}{3}n-O(1)$ and $\hat{R}_2(P_{n}^{(k)})\geq \lceil\log_2(k+1)\rceil n-O_k(1)$ for $k\geq 4$. We consider the problem of giving a lower bound on the $r$-color size-Ramsey number of $P_{n}^{(k)}$ (for fixed $k$ and growing $r$). Our main result is that $\hat{R}_r(P_n^{(k)})=Ω_k(r^kn)$ which generalizes the best known lower bound for graphs mentioned above. One of the key elements of our proof is a determination of the correct order of magnitude of the $r$-color size-Ramsey number of every sufficiently short tight path; i.e. $\hat{R}_r(P_{k+m}^{(k)})=Θ_k(r^m)$ for all $1\leq m\leq k$. All of our results generalize to $\ell$-overlapping $k$-uniform paths $P_{n}^{(k, \ell)}$. In particular we note that when $1\leq \ell\leq \frac{k}{2}$, we have $Ω_k(r^{2}n)=\hat{R}_r(P_{n}^{(k, \ell)})=O((r^2\log r)n)$ which essentially matches the best known bounds for graphs mentioned above. Additionally, in the case $k=3$, $\ell=2$, and $r=2$, we give a more precise estimate which implies $\hat{R}_2(P^{(3)}_{n})\geq \frac{28}{9}n-O(1)$, improving on the above-mentioned lower bound of Winter in the case $k=3$.