I'm a bit confused about the question. What if the bipartite graph just consists of $u_1, u_2$ and $v$ with a single edge from $u_1$ to $v$? There is only one maximum matching and it doesn't satisfy the property.

Sorry, I see that it is not true that if m+1 is even, then there is a proper edge coloring of K_{m+1} with m colors in which there is a rainbow path because this already fails when m+1=4. However, I'm still hopeful that it is true for large enough m or perhaps for m of a certain type. Either way, it is known that there is always a very long rainbow path (look up work on rainbow paths in properly edge colored complete graphs) in an even more general setting, so there are certainly infinitely many n for which there is a linear path covering (1-o(1))n vertices of S.

(continued) I couldn't find a simple reference for the fact that every complete graph on m+1 vertices (with m+1 even) has a proper edge coloring with m colors in which there is a rainbow path of length m, but I am fairly confident this is true.

I think it's easy enough to show that there are infinitely many n for which there is an STS on n vertices with a linear path on n vertices. In the "doubling construction" you start with an STS on m vertices, call it S, and a complete graph on m+1 vertices (where m+1 is even). This complete graph has a proper edge coloring with m colors and I claim that it has a proper edge coloring with m colors in which there is a rainbow path of length m. Each vertex of S is associated with one color class from the complete graph and the rainbow path mentioned above gives the desired linear path in S.

I'm not sure which of these will be helpful to you, but I might start by looking at Theorem 1 in "Graphs with many r-cliques have large complete r-partite subgraphs" by Nikiforov, Lemma 2.1 and Theorem 2.2 in "Hypergraph Turan Problems" by Keevash, and this question together with the answers and comments mathoverflow.net/questions/234278/…

Do you have an example of a non complete graph with chromatic number 4 which just satisfies the first property; that is, for all $v\in G$, there exists a coloring $c$ such that $c(u)\neq c(v)$ for all $u\in V(G)$? (For chromatic number 3, an odd cycle of length at least 5 works.)

Unfortunately, you didn't get the answer you were looking for, but I'd just like to point out that the most promising direction to me is the one pointed out by @TimothyChow (my "answer" was weakened by the thing I pointed out in the addendum). That is, to focus on minimal strongly connected digraphs. These are much more complicated than trees, but there are some nice structural results here doi.org/10.1016/j.laa.2017.11.027 which could be helpful. For instance what additional structural properties can be proved under the assumption that there are no paths of length $\Theta(n)$?

@BrendanMcKay I had never heard the term centroidal vertex, so that is good to know. But by "central vertex" I meant it in the way in which it is defined in this question.

@Timothy Chow I think what you're referring to is answered by part (iii) of my extended comment below. That is, every tree T on n vertices has a vertex v such that every component of T-v has order at most n/2. In this case, v will be a central vertex.

One simple comment is that if $m\leq \frac{1}{r}\binom{n}{2}$, then $R((n,m);r)=n$ (by taking the majority color class). So the parameter is only interesting for "dense" graphs.

According to the discussion at the top of page 585 (third page) in Foreman, Matthew, and Andras Hajnal. "A partition relation for successors of large cardinals." Mathematische Annalen 325, no. 3 (2003): 583-623, it looks like ZFC+GCH does not imply $\omega_2\to (\beta)^2_2$ for all ordinals $\beta<\omega_2$. That being said, I admit that I don't fully understand the sentence which follows explaining the reason why.

Thank you for that reference. On page 2, they mention some open problems under the assumption of GHC. What I'm interested in (assuming GHC) is whether $\omega_2\to (\beta)^2_2$ for all ordinals $\beta<\omega_2$? If this has a positive answer, then based on what they say on page 2, I imagine it should be hard to prove. However, if it has a negative answer, perhaps there is already a reference which answers my question?

Is anything stronger known if the final $\kappa$ in the subscript is replaced by a 2? That is, can the $(2^\kappa)^+$ be made smaller, or the $\kappa^+$ be made larger?

Is there a particular reason you are only interested in the locally finite case? I'm not saying I could provide such an example, but would you be interested in an example where $|N^+(x)|$ is infinite, but $|N^{++}(x)|$ is finite, or say $|N^+(x)|$ is uncountable, but $|N^{++}(x)|$ is countable?