Our Approach: 
 

We started with an inductive approach to showing that diagonal Ramsey Numbers are even on 2 colors. For R(k,k) = n, we have a configuration of n-1 connected vertices without any k-cliques of either color. 
 

Starting from this configuration on n-1 vertices (called a draw state as no color has formed a clique), we add a singular connected vertex v and fill in the n-1 new edges connecting v to the previous n-1 vertices with different colors. Since R(k,k) = n, we can use pidgeonhole arguments on the newly added edges to provide insight into how and why a k-clique must form on a bi-edge-colored n-clique. 

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Showing R(k,k) is even:

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We start with some even number m for which an m-clique has a draw state. We can then add a vertex and connect m new edges while attempting to maintain the draw state. If this is shown to be true, then the largest draw state must be odd, meaning that the Ramsey Number R(k,k) must be even. 

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Tianbo's approach: 

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Tianbo's thesis outlines how he built a neural network to fill in the new edges to find a "drawing strategy", which is key in finding out why the k-clique must form on n conneted vertices. I recommend reaching out to him here for more details if you are interested.