Sample compression conjecture (linear bound in VC-dimension)

Prove that every hypergraph with Vapnik–Chervonenkis dimension d admits a sample compression scheme whose size is linear in d, thereby establishing the sample compression conjecture.

Background

Littlestone and Warmuth showed that bounded-size sample compression implies bounded VC-dimension, and Moran and Yehudayoff proved that any class of VC-dimension d admits a (improper) sample compression scheme of size 2^{O(d)}. However, a linear bound in d is the central conjecture posed by Floyd and Warmuth, with strong evidence of optimality: every class of VC-dimension d needs size at least d/5 and some classes require at least d.

Despite decades of effort, only exponential-in-d upper bounds are known in general. This paper focuses on special graph-induced hypergraphs (balls in graphs) and obtains nearly tight bounds in several structured settings, but the general conjecture remains unresolved.