There is no bound on Borel classes of the graphs in the Luzin-Novikov theorem
We show that for every ordinal $α\in [1, ω_1)$ there is a closed set $F \subset 2^ω\times ω^ω$ such that for every $x \in 2^ω$ the section $\{y\in ω^ω; (x,y) \in F\}$ is a two-point set and $F$ cannot be covered by countably many graphs $B(n) \subset 2^ω\times ω^ω$ of functions of the variable $x \in 2^ω$ such that each $B(n)$ is in the additive Borel class $\boldsymbol Σ^0_α$. 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 $Π^0_1$ set in $ω^ω$ 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 $σ$-compact sections.