Pavle Blagojevic, Benjamin Matschke, and Guenter Ziegler settled the “colorful Tverberg’s conjecture.” (Problem 6 in this post.) This gives a sharp version for Zivaljevic and Vrecica theorem, and crossed the “connectivity of chessboard complexes barrier”. Here is the link to the breakthrough paper.
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I think this is the new version:
http://front.math.ucdavis.edu/0911.2692
I think this is already a follow-up paper with further results
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