What are largest betti numbers $b_2$ and $b_3$ of threedimensional CalabiYau manifolds that are discovered for today?
Is there some nice reference?
What are largest betti numbers $b_2$ and $b_3$ of threedimensional CalabiYau manifolds that are discovered for today? Is there some nice reference? 


Since mirror symmetry exchanges the Hodge numbers $h^{1,1} = b_2$ and $h^{2,1} = \frac{1}{2}(b_3  1)$, it is perhaps more natural (and of course equivalent) to discuss these. The recordholders all come from the list of hypersurfaces in toric fourfolds, constructed by Kreuzer and Skarke. The largest value of $h^{1,1}$ is $491$, and the same for $h^{2,1}$. These two manifolds (which are mirror) also hold the record for largest $h^{1,1} + h^{2,1}$, which is 502. There is a third manifold which shares this record; its Hodge numbers are $(251,251)$, and it is also in the KreuzerSkarke list. See this recent paper and this one for nice plots and a discussion of these CalabiYau threefolds and others with large Hodge numbers. 

