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3. Extending the construction in Fig. VI.2, find for each t 2 a t-regular graph that shows that R(3, t + 1) > 3t - 1. FIGURE VI.2. A graph showing that R(3, 4) > 8.

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We start with considering figure VI.2 which is a 3-regular graph in which all the vertices are placed equidistantly on a circle, each vertex is connected by black edges only to the 3 vertices most opposite to it, and the rest of the vertices of the complete graph can be imagined to be connected by red color (in the picture we have shown 3 of the red edges for the discussion below, but we can imagine all the rest of the red edges). By construction, there is no black 3-clique but there are red 3-cliques. We can see that no red 4-clique is possible by considering any red 3-clique, as follows: By construction, a red 3-clique can only be formed by 3 consecutive vertices. Between them at least one of these 3 vertices is connected by black to any vertex that is not in this red 3-clique. This is because there are only 3 - 1 = 2 vertices on each side of any vertex that are not connected to it by black so, given...

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