Lower bounds for the covering numbers C(24,14,4) and C(25,15,5)
A Lean proof that every family of 14-element subsets of a 24-element set covering all 4-element subsets has at least 20 members, so the covering number C(24,14,4) is at least 20, improving the previously tabulated lower bound of 19. Point derivation (the Schönheim bound) then gives C(25,15,5) >= 34, improving the tabulated 32. The proof excludes a hypothetical 19-block covering: a sum-of-squares identity for the minimum-degree incidence vectors forces balanced components, a rigidity theorem for eleven-block (22,12,3) covers controls suitable completions and derivatives, and two counting…
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