First we focus on the size-Ramsey number of a path $P_n$ on $n$ vertices. In particular, we show that $5n/2 - 15/2 \le \hat{r}(P_n)\le 74n$ for $n$ sufficiently large. This improves the previous lower bound due to Bollobás, and the upper bound due to Letzter. Next we study long monochromatic paths in edge-colored random graph $G(n,p)$ with $pn\to\infty$. Recently, Letzter showed that a.a.s. any 2-edge coloring of $G(n, p)$ yields a monochromatic path of length $(2/3-o(1))n$, which is optimal. Extending this result, we show that a.a.s. any 3-edge coloring of $G(n, p)$ yields a monochromatic path of length $(1/2-o(1))n$, which is also optimal. We will also discuss this problem for an arbitrary number of colors. We also consider a related problem and show that for any $r\ge 2$, a.a.s. any $r$-edge coloring of $G(n, p)$ yields a monochromatic connected subgraph on $(1/(r-1)-o(1))n$ vertices, which is also tight. Finally, we discuss some extensions of the above results for random hypergraphs.

This is a joint work with Paweł Prałat and also with Patrick Bennett, Louis DeBiasio, and Sean English.