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arXiv · 2607.23766

The covering number C(12, 6, 4) is 41

Charlie Krug
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Abstract

A $t$-$(v,k,\lambda)$ covering is a collection of $k$-subsets (blocks) of a $v$-set such that every $t$-subset of points lies in at least $\lambda$ blocks; the covering number $C_\lambda(v,k,t)$ is the least number of blocks in such a collection, and one writes $C(v,k,t)$ when $\lambda=1$. The recorded bounds for $C(12,6,4)$ have been $40 \le C(12,6,4) \le 41$. We show that no $4$-$(12,6,1)$ covering with $40$ blocks exists, and hence that $C(12,6,4)=41$. A counting argument shows that in a hypothetical $40$-block covering every point lies in exactly $20$ blocks, the link of every point is an optimal $3$-$(11,5,1)$ covering with a forced degree sequence, and the six pairs of points of degree $10$ form a perfect matching; an exhaustive case analysis over the orbits of a group of order $3840$, carried out by satisfiability solving, then shows that no optimal $3$-$(11,5,1)$ covering occurs as such a link. Each of the $81$ formulas in the primary proof has an unsatisfiability certificate checked by drat-trim and by the formally verified checker cake_lpr; two additional cross-encoding certificates are checked by the same pipeline. The lower-bound argument uses no tabulated covering number: its only numerical input, $C(10,4,2) \ge 9$, is itself certified. As a by-product the certificates yield a self-contained certified proof that the optimal $3$-$(11,5,1)$ covering is unique up to isomorphism. Equivalently, the Tur\'an number $T(12,8,6)$ is $41$; the new value propagates to improved lower bounds for $C(13,7,5)$, $C(14,8,6)$, $C(15,9,7)$ and $C(16,10,8)$.

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Charlie Krug. 2026-07-26. The covering number C(12, 6, 4) is 41. https://arxiv.org/abs/2607.23766

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