We prove that if $G=(V,E)$ is an $\omega$-stable (respectively, superstable) graph with $\chi (G)>\aleph _0$ $2^{\aleph _0}$) then $G$ contains all the finite subgraphs of shift $\mathrm {Sh}_n(\omega )$ for some $n$. a variant this theorem graphs interpretable in stationary stable theories. Furthermore, $\operatorname {U}(G)\leq 2$ we $n\leq suffices.

In an earlier paper, the present authors (2013) [1] introduced the alternating chromatic number for hypergraphs and used Tucker’s Lemma, an equivalent combinatorial version of the Borsuk-Ulam Theorem, to show that the alternating chromatic number is a lower bound for the chromatic number. In this paper, we determine the chromatic number of some families of graphs by specifying their alternating...

We introduce joins of signed graphs and explore the chromatic number all-positive all-negative joins. prove an analogue to theorem that join two equals sum their numbers. Given graphs, is usually less than numbers, by amount depends on new concept deficiency a signed-graph coloration.