An upper bound for a ramsey type problem for k-connected subgraphs - Chen, Murong; Xie, Qiqin (Graphs and combinatorics, 2025)
For any positive integer k , let r 2 ( k ) denote the smallest integer n such that every 2-edge-colored complete graph K n contains a monochromatic k -connected subgraph. Matula established the bound 4 ( k - 1 ) + 1 ≤ r 2 ( k ) < ( 3 + 11 / 3 ) ( k - 1 ) + 1 . It is known that r 2 ( k ) = 4 ( k - 1 ) + 1 f o r k = 1 , 2 (by Bollobás and Gyárfás) and for k = 3 (by Liu, Morris, and Prince). We prove that for k ≥ 2 and n > ( 3 + 497 - 1 16 ) ( k - 1 ) , every 2-edge-colored K n contains a monochromatic k -connected subgraph with at least 2 ( k - 1 ) vertices. This result improves the upper bound of r 2 ( k ) to ⌈ ( 3 + 497 - 1 16 ) ( k - 1 ) ⌉ for all k ≥ 4 .