On induced Ramsey numbers

Induced Ramsey-type theorems

“…A red-blue edge-coloring of the edges of a graph partitions the graph into two monochromatic subgraphs, the red graph, which contains all vertices and all red edges, and the blue graph, which contains all vertices and all blue edges. The weak induced Ramsey number r weak ind (H 1 , H 2 ), introduced by Gorgol and Luczak [33], is the least positive integer n such that there is a graph G on n vertices such that for every red-blue coloring of the edges of G, either the red graph contains H 1 as an induced subgraph or the blue graph contains H 2 as an induced subgraph. Note that this definition is a relaxation of the induced Ramsey numbers since we allow blue edges between the vertices of red copy of H 1 or red edges between the vertices of blue copy of H 2 .…”

On induced Ramsey numbers for multiple copies of graphs

“…For any ‐vertex graph , . Gorgol and Łuczak [18] obtained the exact value of induced Ramsey number for a matching vs a complete graph and Grünewald for two matchings [19].…”

“…In Theorem 10 we prove that , for which shows the tightness of the lower bound. For the upper bound it is proved that [18]. …”

On lower bound for induced Ramsey numbers

“…At the end we mention that the only known exact values (not concerning the pairs of small graphs) are for a pair of stars by Harary, Nešetřil and Rödl [12], matching versus complete graphs by Gorgol and Luczak [11] and for stars versus complete graphs by Gorgol [10]. The two latter will serve as examples of sharpness of our theorems.…”

“…It is worth to notice that if we allow the graph G not to be connected this lower bound is sharp. Gorgol and Luczak [11] showed the exact value of the induced Ramsey number for a matching and a complete graph.…”