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We present a model of set theory, in which, for a given $n\ge2$, there existsa non-ROD-uniformizable planar lightface $\varPi^1_n$ set in $\mathbbR\times\mathbb R$, whose all vertical cross-sections are countable sets (and infact Vitali classes), while all planar boldface $\bf\Sigma^1_n$ sets withcountable cross-sections are $\bf\Delta^1_{n+1}$-uniformizable. Thus it is truein this model, that the ROD-uniformization principle for sets with countablecross-sections first fails precisely at a given projective level.

Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy