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

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.

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.