The Ramsey number $R(3, 3, \ldots, 3)$, where there are $c$ copies of $3$, represents the minimum natural number $n$ such that if the edges of $K_n$ are colored using one of $c$ colors, the graph necessarily contains a monochrome triangle. For example, the case of $R(3, 3) = 6$ is sometimes called the Theorem on Friends and Strangers: any group of six people contains either three mutual friends or three mutual strangers (or both). We also know $R(3, 3, 3) = 17$: if the edges of $K_{17}$ are assigned one of three colors, then a monochrome triangle must exist somewhere in the graph.
There’s a standard induction proof that shows that $R(3, 3, \ldots, 3) \le \lceil c! \cdot e\rceil$. It’s a generalization of the famous proof of the Theorem on Friends and Strangers, and specifically arises because the following expression gives the number of nodes needed to force a monochrome triangle:
$$c!\left(\frac{1}{0!} + \frac{1}{1!} + \dots + \frac{1}{c!}\right) + 1\text.$$
The number $\lceil c! \cdot e\rceil$ is closely related to another expression: the number of paths between a pair of distinct nodes in the graph $K_{n}$, which is $\lfloor (n-2)! \cdot e\rfloor$. This follows because the number of such paths is counted by the expression
$$(n-2)!\left(\frac{1}{0!} + \frac{1}{1!} + \dots + \frac{1}{(n-2)!}\right)\text.$$
Given that these are nearly the same formula, it seems like there might be a way to link the two concepts together. Specifically, it seems like the number of nodes needed in a complete graph to force a monochrome triangle using $c$ colors should be somehow related to the number of paths between a pair of distinct nodes in a $(c+2)$-clique.
I’ve been thinking about how to link these, but I’m completely lost about how to do so because these two problems seem so different - one involves counting all objects of some type in an uncolored graph, and one involves proving the existence of a small object regardless of how an object is colored.
Is there a known connection between this upper bound on Ramsey numbers and paths between distinct nodes in a clique?