, Volume 5, Issue 2, pp 137-139

The chromatic number of the product of two ℵ1-chromatic graphs can be countable

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Abstract

We prove (in ZFC) that for every infinite cardinal ϰ there are two graphsG 0,G 1 with χ(G 0)=χ(G 1)=ϰ+ and χ(G 0×G 1)=ϰ. We also prove a result from the other direction. If χ(G 0)≧≧ℵ0 and χ(G 1)=k<ω, then χ(G 0×G 1)=k.