Asymptotic bounds for some bipartite graph: Complete graph Ramsey numbers
The authors define a variation on the usual Ramsey numbers: \(R_c(P^{\text{conv}}_e,t)\) is defined as the smallest integer \(n\) so that, if the vertices of the complete graph are cyclically ordered and its edges are \(t\)-colored, there exists a monochromatic path of length \(l\) which respects the order. \(R_c(C^{\text{conv}}_k, C_e^{\text{conv}})\) is similarly defined where \(C^{\text{conv}}_e\) refers to circuits for length \(l\) which respect the ordering. The authors prove several results including: Theorem 2.2. For positive integers \(l\) and \(t\), \(1+ l(l- 1)^{t-1}\leq R_c(P^{\text{conv}}_l,t)\leq 1+l^t\). Theorem 3.3. For positive integers \(l\) and \(k\), \[ (k-1)(l- 1)+ 1\leq R_c(C^{\text{conv}}_k, C^{\text{conv}}_l)\leq 2kl- 3k- 3l+ 6. \]
scientific article; zbMATH DE number 1462272