A combinatorial property of cardinals (Q2781310)

For a nonempty set \(X\), let \([X]^{\leq 2}\) be the set of all subsets of \(X\) with at most two elements, and \([X]^{<\omega}\) the set of all finite subsets of \(X\). Let a two-coloring \(F:[X]^2\to \{0,1\}\) be given. For a mapping \(f: X\to [X]^{<\omega}\), a pair \((x,y)\in X\times X\) is said to be \(0\)-pair if \(F(x,t)= 0\) for every \(t\in f(y)\), and \(F(u,y)= 0\) for every \(u\in f(x)\). The definition of \(1\)-pairs is analogous. For an infinite cardinal \(\kappa\) let \(P(\kappa)\) denote the following statement: There exists a function \(F:[\kappa]^{\leq 2}\to \{0,1\}\) such that for every mapping \(f: \kappa\to [\kappa]^{<\omega}\) there exists a \(0\)-pair and a \(1\)-pair. Then it is proved: 1) \(P(\kappa)\) fails for \(\kappa\leq\omega_1\). 2) Using GCH, \(P(\kappa)\) holds for every \(\kappa\geq \omega_2\). 3) Using ZFC there follows: \(P(\kappa)\) holds if \(\kappa^{\aleph_2}= \kappa\). 4) Using GCH, the following is proved: Assume that \(\text{cf}(\lambda)> \omega_1\). Then it is consistent that \(2^{\aleph_1}= \lambda\) and \(P(\omega_2)\) holds.