Determine the hardest qrel matrix under sign-rank-based limits

Determine, for fixed numbers of documents and top-k relevant documents per query, which binary query relevance (qrel) matrix is provably the hardest to represent for single-vector embedding models—formally, which matrices require the largest embedding dimension or maximize sign-rank—and provide a definitive theoretical proof identifying such matrices.

Background

In constructing the LIMIT dataset, the authors aimed to stress-test single-vector models by instantiating qrel matrices that are hard to represent. They conjecture that more interconnected (denser) qrel patterns are harder but acknowledge the difficulty of proving hardness due to the notorious difficulty of establishing sign-rank for specific matrices.

They explicitly state they could not prove which qrel matrix is hardest, leaving a concrete theoretical identification and proof as an open task.