Let \(R\subseteq\omega \times\omega \) be a binary relation. \(f: \omega\to \omega\) almost \(\times\)-sections R if \(\omega\vDash (\exists n)(\forall x>n)R(x,f(x)). <\xi >\) is the RET of \(\xi \subseteq\omega \). Define \([\alpha,\beta]=\{\xi| \alpha\subseteq \xi\subseteq \alpha\cup \beta\}, [\alpha,\beta]^{<\omega}\) be the set of elements of [\(\alpha\),\(\beta]\) which are finite and \([\alpha,\beta]^{\omega}\) be the set of elements of [\(\alpha\),\(\beta]\) which are infinite; \([\beta]^{<\omega}\) be \([\emptyset,\beta]^{<\omega}\). A set \(P\subseteq [\omega]^{\omega}\) is porous if for every \(\alpha\in [\omega]^{<\omega}\) and \(\beta\in [\omega]^{\omega}\) there is a \(\xi\in [\alpha,\beta]^{\omega}\) such that \([\alpha,\xi]^{\omega}\) is disjoint with P.
In this paper the author proves: Proposition. \(P\subseteq [\omega]^{\omega}\) is porous if and only if it is meager. Theorem 1. If R is a binary relation which is not almost \(\times\)-sectioned by an almost combinatorial function then \(\{\xi| \Lambda\vDash (\exists y)R(<\xi >,y)\}\) is (i) porous, and (ii) null. Theorem 2. If R is a binary relation, \(Y\in\Lambda \), and \(\{n\in\omega |\Lambda \vDash\neg R(n,Y)\}\) is infinite then \(\{\xi| \Lambda\vDash R(<\xi >,Y)\}\) is porous and null. - ”Null” means of measure zero in the product measure on \(\{0,1\}^{\omega}\) induced by the unbiased probability measure on \(\{\) 0,1\(\}\).