Large entire cross-sections of second category sets in Rn+1

By the Kuratowski–Ulam theorem, if A⊆Rn+1=Rn×R is a Borel set which has second category intersection with every ball (i.e., is “everywhere second category”), then there is a y∈R such that the section A∩(Rn×{y}) is everywhere second category in Rn×{y}. If A is not Borel, then there may not exist a large cross-section through A, even if the section does not have to be flat. For example, a variation on a result of T. Bartoszynski and L. Halbeisen shows that there is an everywhere second category set A⊆Rn+1 such that for any polynomial p in n variables, A∩graph(p) is finite. It is a classical result that under the Continuum Hypothesis, there is an everywhere second category set L in Rn+1 which has only countably many points in any first category set. In particular, L∩graph(f) is countable for any continuous function . We prove that it is relatively consistent with ZFC that for any everywhere second category set A in Rn+1, there is a function which is the restriction to Rn of an entire function on Cn and is such that, relative to graph(f), the set A∩graph(f) is everywhere second category. For any collection of less than ℵ02 sets A, the function f can be chosen to work for all sets A in the collection simultaneously. Moreover, given a nonnegative integer k, a function of class Ck and a positive continuous function , we may choose f so that for all multiindices α of order at most k and for all x∈Rn, |Dαf(x)−Dαg(x)|<ε(x). The method builds on fundamental work of K. Ciesielski and S. Shelah which provides, for everywhere second category sets in ω2×ω2, large sections which are the graphs of homeomorphisms of ω2. K. Ciesielski and T. Natkaniec adapted the Ciesielski–Shelah result for subsets of R×R and proved the existence in this setting of large sections which are increasing homeomorphisms of R. The technique used in this paper extends to functions of several variables an approach developed for functions of a single variable in previous related work of the author.