Erdős 699: current results and remaining questions

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For every $1\le i<j\le n/2$, must a prime $p\ge i$ divide both $\binom ni$ and $\binom nj$? A bad triple is an admissible triple for which no such prime exists. The threshold is $p\ge i$, including $p=i$ when $i$ is prime.

The problem remains open. Van Doorn and Rocca's retained, unpublished manuscripts establish the fixed-index finiteness theorem for $i\ge4$, settle $i=1,2$, and exclude a uniform range of large indices.

Reading guide

These are the main entry points.

1. Problem statement and sources. The canonical statement, bibliography, and current status.
2. Van Doorn and Rocca's argument. A theorem-by-theorem account of the retained manuscript, with its full text: the local congruences, polynomial divisibility, discriminant bound, and the ineffective finiteness theorem.