The covering number for category and partition relations on \(P_\omega(\lambda)\) (Q2773369)

The cardinal \(\text{cov}(M)\), the minimal cardinality of a family of meager sets of reals whose union covers the whole real line, has many interesting combinatorial guises. The author investigates, following earlier works, its connection with partition relations. One of the main results is the following. Fix an infinite cardinal \(\lambda\), and consider ideals \(J\) whose domain is \(P_\omega(\lambda)\), the collection of finite subsets of \(\lambda\). Then \(J^+\) denotes \(\{A\subseteq P_\omega(\lambda) : A\not\in J\}\), and the partition property NEWLINE\[NEWLINEJ^+\overset{\omega}{\to} (J^+)^2NEWLINE\]NEWLINE asserts that for each \(A\in J^+\) and each coloring \(F:\omega\times P_\omega(\lambda)\to \{0,1\}\), there exists \(B\subseteq A\) such that \(B\in J^+\) and all points \((\max(a\cap\omega),b)\) -- where \(a,b\in B\) and \(\max(a\cap\omega)<\max(b\cap\omega)\) -- have the same color. Finally, let \(I_{\omega,\lambda}\) denote the ideal of all \(A\subseteq P_\omega(\lambda)\) such that for some \(a\in P_\omega(\lambda)\) no member of \(A\) contains \(a\). When \(\lambda=\omega\), \(I_{\omega,\lambda}^+\overset{\omega}{\to} (I_{\omega,\lambda}^+)^2\) is just a reformulation of Ramsey's Theorem. Thus a natural question is whether, and to what extent, this property remains true for larger \(\lambda\). The author proves that it is true if, and only if, \(\lambda<\text{cov}(M)\). There are a handful of related results in the paper, to which the reader is referred for more details; the paper is self contained.