We show that for every countable ordinal alpha > 0 there is a closed subset F of the product of the Cantor space C and the Baire space N such that for every x from C the section of F at x is a two-point subset of N and F cannot be covered by countably many graphs B(n)'s of functions from C to N such that each B(n) is in the additive Borel class alpha.
This rules out the possibility to have a quantitative version of the Luzin-Novikov theorem. The construction is a modification of the method of Harrington who invented it to show that there exists a countable effectively closed set in N containing a non-arithmetic singleton. By another application of the same method we get closed sets excluding a quantitative version of the Saint~Raymond theorem on Borel sets with sigma-compact sections. This is a joint work with Petr Holický.