On the limit superior of analytic sets (Q1061122)

\textit{M. Laczkovich} proved [Anal. Math. 3, 199-206 (1977; Zbl 0362.04001)] that if the sets \(A_ 0,A_ 1,..\). are Borel sets of reals such that for every infinite subsequence H, lim sup\(\{\) \(A_ j: j\in H\}\) is uncountable, then for an appropriate infinite subsequence H even the intersection of the \(A_ j's\) (j\(\in H)\) is uncountable. He also showed that under the continuum hypothesis there are (non-Borel) sets for which this does not hold. In this paper we extend the first result to analytic sets (in Polish spaces), show that under \(MA(\omega_ 1)\) it is true for any sets, but can be false with the cardinality of continuum 'anything'. Under the axiom of constructibility the statement is false for co-analytic sets. At the end of the paper a different proof for the analytic case is given: from the \(MA(\omega_ 1)\) result using absoluteness.