An uncountably chromatic triple system
A triple system is a pair \((V,E)\) with \(E \subseteq [V]^3\). Its chromatic number \(\text{Chr}(V,E)\) is the least cardinal \(\mu\) such that there exists a function \(f : V \rightarrow \mu\) such that for each \(\{ x,y,z \} \in E\) we have \(| \{ f(x), f(y), f(z) \}| \geq 2\). For \(3 \leq n < \omega\), \(\mathcal{C}_n\) is the triple system \((V,E)\) with \(V = \{ x_i, y_i : i<n \}\) and \(E = \{\{x_i,y_i,x_{i+1} \} : i<n \}\) (with \(x_n = x_0\)), and \(\mathcal{T}_0\) is the triple system with vertex set \(\{ x,y,z,t\}\) and triples \(\{ x,y,z\}\), \(\{x,y,t\}\). In an earlier paper, \textit{A. Hajnal} and \textit{P. Komjáth} [Acta Math. Hung. 119, No. 1--2, 1--13 (2008; Zbl 1199.03029)] showed: If an uncountably chromatic triple system omits \(\mathcal{T}_0\), then it contains all odd circuits of length \(\geq 7\). Here the author shows that it is consistent that GCH holds and there is a triple system \(\mathcal {H}\) with \(| \mathcal {H}| = \aleph_2\), \(\text{Chr}(\mathcal {H}) = \aleph_1\) and \(\mathcal {H}\) does not contain \(\mathcal {T}_0\), \(\mathcal {C}_3\) or \(\mathcal {C}_5\).
scientific article; zbMATH DE number 5651918
scientific article