is there a better upper bound for the Ramsey-Number for monochromatic paths with more than two colors?

Question

Define $\rho(k,l)$ where $k \ge l$ to be $R(P_k,P_l)$, Which is the smallest number $R$ for which an $R$ clique whose edges are colored red or blue, is guaranteed to contain a red $P_k$ or a blue $P_l$. in this paper, Ramsey the authors show that $\rho(k,l) = k + \lceil (l+1)/2 \rceil$.

Now define $\delta (n,t) = R(P_t,P_t, \ _{n-times}..., P_t)$. Which is the smallest number $\delta$ for which any coloring of the edges of a $\delta$ -clique with $n$ colors contains a monochromatic path with $t-1$ edges. The familiar upper bound of $\delta(n,t) \le (t-1)^n +1$ is discussed here (Ramsey number for paths).

However, $\rho(t-1,t-1) = \delta(2,t) = t-1 + \lceil(t/2)\rceil$ which is linear in $t$. I wonder if somehow this could be a starting point for a better upper bound for $\delta(n,t)$.In particular, are the better known upper bounds in the literature? Thank you in advance.

Answer 1

Much better upper bounds are easy to prove. We have $\delta(n,t) \le nt$ by the following argument:

1. In a $n$-edge-coloring of $K_{nt}$, there is a color class with at least $\frac1n \binom{nt}{2}$ edges - and therefore average degree $\frac{nt-1}{n} = t - \frac1n$.
2. By the Erdős-Gallai theorem, a graph with average degree more than $t-2$ contains a $t$-vertex path.

The last time I checked, the state on $\delta(n,t)$ was that it's at least $(n-1 + o(1))t$ (source) and at most $(n-\frac12+o(1))t$ (source), but these are both harder to obtain.