The Covering Number $C(12,6,4)$
Prior state unknown→proved
closes a one-block gap in the covering tables; the analogous next case is not reachable by this method
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combinatorics / Design theory
The Covering Number $C(12,6,4)$
A $t$-$(v,k,\lambda)$ covering is a family of $k$-subsets of a $v$-set meeting every $t$-subset at least $\lambda$ times, and $C(v,k,t)$ is the least number of blocks. The recorded bounds for $C(12,6,4)$ were $40 \le C(12,6,4) \le 41$. No $4$-$(12,6,1)$ covering with $40$ blocks exists, so $C(12,6,4) = 41$.
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A $t$-$(v,k,\lambda)$ covering is a family of $k$-subsets of a $v$-set meeting every $t$-subset at least $\lambda$ times, and $C(v,k,t)$ is the least number of blocks. The recorded bounds for $C(12,6,4)$ were $40 \le C(12,6,4) \le 41$. No $4$-$(12,6,1)$ covering with $40$ blocks exists, so $C(12,6,4) = 41$.
closes a one-block gap in the covering tables; the analogous next case is not reachable by this method
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