Erdős, Pál; Hajnal, András Some remarks on set theory. VIII. (English) Zbl 0095.03902 Mich. Math. J. 7, 187-191 (1960). The authors consider independent sets and graphs (cf. also Erdős-Fodor, Zbl 0078.04203). Let \(R\) denote the set of real numbers; for every \(x \in R\) let \(S(x)\) be such that \(x \not\in S(x) \subset R\). A subset \(S \subset R\) is independent provided for every \(x,y \in S\), \(x \neq y\) one has \(x \not\in S(y)\), \(y \not\in S(x)\). Let \(H_0\) denote the statement: \(R\) can be well-ordered into a \(\Omega_c\)-sequence such that every set which is not cofinal with \(\Omega_c\) has measure 0. Theorem 1: If \(S(x)\) \((x \in R)\) is of measure 0 and is not everywhere dense, there exist 2 real independent numbers \(x \neq y\) (under \(H_0\) there are no 3 independent real numbers). Theorem 2: If \(S(x)\) is bounded and has the exterior measure \(\leq 1\), then there are \(n\) independent real numbers, for every \(1 < n < \omega_0\). A \(\sigma\)-ideal \(I\) of subsets \(R\) is said to have the property \(P\), symbolically \(I \in P\), provided it contains a transfinite sequence \(B_\beta\) \((\beta < \Omega_c)\) of members such that every member of \(I\) is contained in some \(B_\beta\). Theorem 3: If \(\aleph_1 = c\) and \(I \in P\), then each graph \(G_R\) on \(R\) contains an infinite chain or an antichain that is not in \(I\) (the statement may not hold provided \(I\not\in P)\). Theorem 5: Let \(m < c\). Let \(I_\alpha\) (\(\alpha < \Omega_c\)) be a sequence of \(\sigma\)-ideals of subsets of \(R\), each with property \(P\). Then every graph \(G_R\) contains, for every \(n< \omega\), a subgraph \(\{x_i\} \cup \{y_\nu \}\) \((1 < i \leq n\), \(1 < \alpha < \Omega_c\)) such that (\(x_i, y_\alpha\)) is connected or there is an antichain in \(G_R\) which is contained in no \(I_\alpha\). The authors ask whether theorem 5 holds for \(m=c\); they conjecture also that theorem 5 may not hold if the property \(P\) is delated, even for \(n=m=2\). Reviewer: G.Kurepa Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 3 Documents MSC: 03E99 Set theory Keywords:set theory Citations:Zbl 0078.04203 PDFBibTeX XMLCite \textit{P. Erdős} and \textit{A. Hajnal}, Mich. Math. J. 7, 187--191 (1960; Zbl 0095.03902) Full Text: DOI