Lower bounds for finite covering-design numbers
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Let C(v,k,t) be the fewest k-subsets of a v-point set needed to cover every t-subset. Is a 19-block (24,14,4) covering possible, and what bound follows for C(25,15,5)?
Why it matters and what counts as progress
Why it matters: A certified impossibility improves covering-design tables and can propagate to larger parameters through point derivation.
What counts as progress: Rule out a smaller covering by an exact certificate, construct a matching covering or propagate improved lower bounds through point derivation.
How the problem developed
- 12 Oct 2026 · Formalized lower bounds [1]
James Chang Registered a Lean-verified proof of C(24,14,4)≥20, which by the Schönheim bound yields C(25,15,5)≥34, improving the cited tabulated lower bounds.
AI contribution
Lower bounds for the covering numbers C(24,14,4) and C(25,15,5)
Every family of 14-subsets of a 24-point set that covers every 4-subset has at least 20 blocks, so C(24,14,4)≥20. The Schönheim point-derivation bound then gives C(25,15,5)≥34.
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