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Aug 24, 2014 at 22:04 comment added Pablo I think that it is rather apparent that there is a transversal of size $q+1$ which is smaller than $2k$ for large enough $k$. In your suggestion, you only use the independence of $A$ and $B$ in bounding their size which is clearly too coarse.
Aug 24, 2014 at 19:58 history answered Zack Wolske CC BY-SA 3.0