2020/11/27 by Fang, Chunqiu, Győri, Ervin, Xiao, Chuanqi +1
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2011.13715
We call a 4-cycle in K_n1, n2, n3 multipartite, denoted by C4multi, if it contains at least one vertex in each part of K_n1, n2, n3. The Turán number ex(K_n1,n2,n3, C4multi) ( respectively, ex(K_n1,n2,n3,\C3, C4multi\) ) is the maximum number of edges in a graph G⊆ K_n1,n2,n3 such that G contains no C4multi ( respectively, G contains neither C3 nor C4multi ). We call a Cmulti4 rainbow if all four edges of it have different colors. The ant-Ramsey number ar(K_n1,n2,n3, C4multi) is the maximum number of colors in an edge-colored of K_n1,n2,n3 with no rainbow C4multi. In this paper, we determine that ex(K_n1,n2,n3, C4multi)=n1n2+2n3 and ar(K_n1,n2,n3, C4multi)=ex(K_n1,n2,n3, \C3, C4multi\)+1=n1n2+n3+1, where n1≥ n2≥ n3≥ 1.