| Given a graph $F$ and a positive integer $r$,
the size Ramsey number $\widehat{R}({F},r)$
is defined as the smallest integer $m$
such that there exists a graph
$G$ with $m$ edges where every $r$-color edge
coloring of $G$ results in a monochromatic copy of $F$.
Let $P_n$ and $C_n$ represent a path and a cycle on
$n$ vertices, respectively. In this paper,
we establish that for sufficiently large $n$,
$\widehat{R}(P_n,P_n,P_n)<772n$.
Furthermore, we demonstrate that for sufficiently large even integers
$n$, $\widehat{R}(P_n,P_n,C_n)\leq 17093n$. For sufficiently large odd integer $n$, we show that $\widehat{R}(P_n,P_n,C_n)\geq (7.5-o(1))n$. |
