Products of Luzin-type sets with combinatorial properties

A topological space is Menger if for any sequence U1, U2, … of open covers of the space, there are finite subsets F1, F2, … of U1, U2, …, respectively, such that the union of F1, F2, … covers the space. If we can request that the sets F1, F2, … are singletons, then the space is Rothberger. A subset of the real line, of cardinality at least cov(M), is cov(M)-Luzin if its intersection with any meager subset of the real line has cardinality strictly smaller than cov(M). Each cov(M)-Luzin set is Rothberger, and thus Menger. Assuming cov(M)=c, Bartoszyński, Shelah, and Tsaban constructed two cov(M)-Luzin sets whose all finite powers are Rothberger but its product space is not Menger (Journal of Symbolic Logic 68 (2003), 1254—1260). We show that such sets exist, assuming cov(M)=cof(M) + cov(M) is regular. Our proof, in contrast to the topological construction of Bartoszyński, Shelah, and Tsaban, is purely combinatorial. We apply this result to local properties of function spaces with pointwise convergence topology. This is a joint work with Grzegorz Wiśniewski.