Question

Can someone explain to me how to calculate R(3,3)= 6? Thanks

Theorem Each of 17 students talked with every other student. They all talked about three different topics. Each pair of students talked about one topic. Prove that there are three students that talked about the same topic among themselves. Proof Denote the topics discussed by the students T1, T2, T3. Consider one of the students, say A. Since A talked about three topics to 16 other students, by the Pigeonhole Principle, there are at least 6 students with whom A talked of a single topic, say, T1. Then there are just to possibilities. If any pair from the 6 students discussed topic T1 we are done. Otherwise, there are 6 students that discussed between themselves only 2 topics - T2 or T3. So we are looking at the number R(3, 3) which is 6; and we are done in this case also. To prove that indeed R(3) = 17 requires to exhibit a coloring of K16 in three colors with no monochromatic triangle. Such a graph exists and is formed by three overlapping copies of the Clebsch graph, each of which is colored in one of the three given colors.

Answer 1

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