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

Abstract

A $t$-$(v,k,λ)$ covering is a collection of $k$-subsets (blocks) of a $v$-set such that every $t$-subset of points lies in at least $λ$ blocks; the covering number $C_λ(v,k,t)$ is the least number of blocks in such a collection, and one writes $C(v,k,t)$ when $λ=1$. 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 Turán 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 $C(16,10,8)$.

Disclosure

“o a weaker forcing regime or a new idea; parameter families where the lifted bound is again tight are the better candidates for a direct replay of this approach. Use of AI assistance During the preparation of this work the author used the large language models GPT-5.6 Sol, GPT-5.6 Terra and Claude Fable 5, in the versions available in July 2026, to assist with exploratory analysis, computational search, supporting code, manuscript drafting and revision, and proofreading. The verification status”

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Classification

Drafting limited passages

Multiplier

5

Verified

2026-08-23T19:07:43.086928Z

Structural counts