Three color Ramsey numbers for graphs with at most 4 vertices

Description

For given graphs H1, H2, H3, the 3-color Ramsey number R(H1, H2, H3) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with 3 colors, then it always contains a monochromatic copy of Hi colored with i, for some 1 6 i 6 3. We study the bounds on 3-color Ramsey numbers R(H1, H2, H3), where Hi is an isolate-free graph different from K2 with at most four vertices, establishing that R(P4, C4, K4) = 14, R(C4, K3, K4−e) = 17, R(C4, K3+e, K4−e) = 17, R(C4, K4− e, K4−e) = 19, 28 6 R(C4, K4−e, K4) 6 36, R(K3, K4−e, K4) 6 41, R(K4−e, K4− e, K4) 6 59 and R(K4−e, K4, K4) 6 113. Also, we prove that R(K3+e, K4−e, K4− e) = R(K3, K4 − e, K4 − e), R(C4, K3 + e, K4) 6 max{R(C4, K3, K4), 29} 6 32, R(K3 +e, K4 −e, K4) 6 max{R(K3, K4 −e, K4), 33} 6 41 and R(K3 +e, K4, K4) 6 max{R(K3, K4, K4), 2R(K3, K3, K4) + 2} 6 79.

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