cs.CCmath.COTheoretical
Upper bounds for the monotone rank of the unique disjointness matrix
Jul 29, 2026
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The paper provides tight bounds for the OR-rank and an upper bound for the SUM-rank of the unique disjointness matrix.
Provided tight bounds for OR-rank and an upper bound for SUM-rank of unique disjointness matrix.
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Abstract
More Like ThisIt is shown that the $\mathsf{OR}$-rank (covering rank) of the $2^n \times 2^n$ unique disjointness matrix is $n^{O(1)}(3/2)^n$, hence the known lower bound $1.5^n$ turns out to be essentially tight. By the way, an upper bound $1.89^n$ is obtained for the $\mathsf{SUM}$-rank (partition rank) of this matrix.