Abstract
AbstractFor every infinite cardinal λ and 2 ≤ n < ω there is a directed graph D of size λ such that D does not contain directed circuits of length ≤n and if its vertices are colored with <λ colors, then there is a monochromatic directed circuit of length n + 1. For every infinite cardinal λ and finite graph X there is a λ‐sized graph Y such that if the vertices of Y are colored with <λ colors, then there is a monochromatic induced copy of X. Further, Y does not contain larger cliques or shorter odd circuits than X. The constructions are using variants of Specker‐type graphs.
Highlights
Joó proved that if n ≥ 2, κ is an infinite cardinal, there is a directed graph (V, D) with no directed circuit of length at most n, such that if V is colored with κ colors, there is a monochromatic directed circuit of length n + 1 [3]
In this note we give such an example using a variant of Specker‐type graphs
In the following part of the paper we prove a similar partition result for undirected graphs
Summary
If X is a finite graph and λ is an infinite cardinal, there is a graph Y such that ∣Y∣ = λ and Y ⇒ (X )1
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