Multicolor bipartite Ramsey numbers for paths, cycles, and stripes

Frank Ramsey introduced the theory that bears his name in 1930. The main subject of the theory are complete graphs whose subgraphs can have some regular properties. Most commonly, we look for monochromatic complete subgraphs, i.e., complete subgraphs in which all of the edges have the same color. Ramsey numbers have attracted the attention of many mathematicians due to their many applications in various fields such as graph theory, geometry, logic, information theory, and number theory. Computing exact values for Ramsey numbers is a rather hard task. A huge amount of computational power is needed to generate all colorings of graphs and check the conditions that should be satisfied by the subgraphs. Given bipartite graphs \(G_1, G_2, \dots, G_t)\), the multicolor bipartite Ramsey number \(BR(G_1, G_2, . . . , G_t)\) is the smallest positive integer \(b\), such that any \(t\)-edge-coloring of \(K_{b,b}\) contains a monochromatic subgraph isomorphic to \(G_i\) colored with the \(i\)-th color for some \(1 \le i \le t\). In the upcoming seminar, I will present my latest results on bipartite Ramsey numbers for paths, cycles, and matchings.