If we have a homogeneous (or min-homogeneous) set of size 20, then certainly we have one of size 4. So: If we prove that for all colorings of n-edges we obtain a (min)homogeneous object of size a+2n+1 then we have certainly obtained one of size at least a, so we may as well assume that a is at least 2n+1. Also using the same idea, given any number z we may as well assume that the least number in the homogeneous set is at least z, since it suffices to obtain a homogeneous set of size z+a to ensure we have a homogeneous set of size a whose least element is at least z.

Let’s see… Ramsey: Let’s say a k-hypergraph is a set A together with a relation on k-subsets of A (k-edges). So a (non-directed) graph (without loops) is just a 2-hypergraph. Is this the usual notation? The size of such a beast is the number of vertices, |A|. It is complete iff each k-edge is in the relation. Then Ramsey’s result becomes: “For all n and k >=1, and all finite sets of colors C, there is an M such that given any coloring f that assigns to each k-edge of the complete k-hypergraph of size M a color in C there is a complete (sub)-k-hypergraph H of size at least n such that all its k-edges have the same coloring.”

For k=2 and 2 colors this can be said in an easier way: For all a there is b such that any (non-directed, without loops) graph in at least b vertices either contains a complete subgraph of size a or an empty subgraph of size a. (Empty meaning there are no edges between the vertices).

“For all n and k >=1, and all finite sets of colors C, there is a M such that for all graphs X of size at least M and all colorings f that assign k edges in X to a color in C, there is a subgraph H of size at least n in X such that all k-sized subgraphs of H have the same coloring.”

The coloring doesn’t make much sense unless we can assign each edge multiple colorings…

Wait, no, but switching them makes the homogeneity definition break… with the switching, this is what I get.

f : {1, …, w}^[k+n] –> {1, …, e + 2}

But |H| = 2n + 1.

Since H’s homogeneity refers to the size of the sets in the domain of f, we then have that “for all A, B in H^[k+n]“… but there are no subsets of H of size k + n (unless, again, n = 0 and k = 2n+1).

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I’m rather confused in general, actually, about the translation from the notation to words. Let’s try Ramsey’s theorem in words… (assume all graphs are connected)

“for all ???s k and subgraph sizes n, and all finite sets of colors C, there is a minimum graph size M such that for all graphs X of size >= M and all colorings f, some subgraph of H of size >= n will be colored the same way twice?”

That’s all wrong. Help anyone?

I can see it both ways, and I can probably think about the hint for Problem 1 more and figure out what these notation mean, but I am a bit confused right now, so I thought I would just ask here.

(Yes, “for each phi in X” )