Two unattainable Schönheim bounds: C(13,5,4) ≥ 150 and C(14,5,4) ≥ 220

# Two unattainable Schönheim bounds: C(13,5,4) ≥ 150 and C(14,5,4) ≥ 220

Ciprian Ioan Macovei — 7 September 2026. Draft for peer review.

Abstract. We prove that coverings of all 4-subsets by 5-subsets require at least 150 blocks on 13 points and at least 220 blocks on 14 points. These improve the lower bounds 149 and 219 in the La Jolla Covering Repository snapshots checked here; the published construction sizes remain 157 and 229. Both proofs use only integer multiplicities and double counting. On 13 points, a pair of minimum multiplicity determines a unique 4-set of excess multiplicity two. Parity and a short classification of the point and pair surpluses then give a contradiction. On 14 points, the three surplus points force one triple of multiplicity twelve, while every 4-set has multiplicity at most two, giving 24 ≤ 22. The proofs require no exhaustive computation. Standard recursion gives eight further improvements over the archived lower bounds. Novelty is qualified by the sources searched.

1. Definitions and the common counting identity

A (v,5,4)-covering is a multiset B of 5-subsets of a v-point set V such that every 4-subset is contained in a block. Let b = |B| and let d(S) count the blocks containing S, with multiplicities. For 0 ≤ |S| ≤ 3, counting the other points of each block gives

Σ_{x∉S} d(S ∪ {x}) = (5 − |S|) d(S). (1)

In particular d(∅) = b and d(Q) ≥ 1 for every 4-set Q. The successive integer lower bounds on triple, pair and point multiplicities are as follows.

For these two values of v, put s_x = d({x}) − m₁ and p_xy = d({x,y}) − m₂. Both are nonnegative integers, and 4m₁ = (v−1)m₂, so

Σ_x s_x = 5b − v m₁, Σ_{y≠x} p_xy = 4s_x. (2)

The sum in one row is at most the sum in all other rows, because the p's are nonnegative and symmetric. Thus s_x ≤ (Σ_y s_y)/2. Call a point ordinary if s_x = 0 and a pair ordinary if p_xy = 0. Every pair incident with an ordinary point is ordinary. All positive pair surpluses are therefore supported on the non-ordinary points.

2. The 13-point bound

Theorem 1. C(13,5,4) ≥ 150.

Assume b = 149. Then Σs_x = 4 and each s_x ≤ 2. For triples T and 4-sets Q write e_T = d(T) − 5 and f_Q = d(Q) − 1. Equation (1), and its sum over triples through a pair P, give

Σ_{Q⊃T} f_Q = 2e_T; (3) Σ_{T⊃P} e_T = Σ_{Q⊃P} f_Q = 2 + 3p_P. (4)

These particular equalities use v = 13. Summing again at a point a gives

Σ_{Q∋a} f_Q = 8 + 4s_a. (5)

Ordinary-pair lemma. An ordinary pair P lies in exactly two triples of positive excess, both with e = 1. If those triples are P ∪ {u} and P ∪ {w}, the only 4-set through P with positive f is Q_P = P ∪ {u,w}, and f_{Q_P} = 2.

To prove the lemma, (4) says that the triple excesses through P sum to two. If one triple P ∪ {u} carried both units, (3) would require a positive 4-set P ∪ {u,z}. Its other triple P ∪ {z} must also have positive excess by (3), a contradiction. There are therefore two triples, each of excess one. Every positive 4-set through P must contain both of their extra points, so it is Q_P. Equation (3) on either triple then gives f_{Q_P} = 2.

It follows that a triple containing an ordinary pair has e ≤ 1, and that a 4-set containing an ordinary pair has f in {0,2}.

The classification is elementary. Let S be the non-ordinary points. Since its positive integers s sum to four and are at most two, their profiles are (2,2), (2,1,1), and (1,1,1,1). Equation (2) determines the pair surpluses in the first two profiles: respectively one edge of weight eight, or two edges of weight four with a common endpoint.

For the last profile, label S = {a,b,c,d}. The four row sums are all four. Comparing the sum of rows a,b with the sum of rows c,d proves p_ab = p_cd; similarly p_ac = p_bd and p_ad = p_bc. Thus opposite edges have weights α, β, γ with α+β+γ = 4. Relabelling permutes these three weights. Their unordered possibilities are

(4,0,0), (3,1,0), (2,2,0), (2,1,1). (6)

Together with the two smaller supports, these are exactly six configurations. No computer enumeration is needed.

First exclude the complete special graph. This is the final possibility in (6), in which every pair in S is special. For any pair P in S, every 4-set through P other than S contains an ordinary point, so its f is even. Equation (4) shows that all six p_P have the same parity, namely f_S modulo two. Each row sums to four and contains three terms of this parity. Hence they are all even. Since they are all positive, each row sums to at least six, a contradiction.

In every remaining configuration, every 4-set contains an ordinary pair: a 4-set outside S contains an ordinary point, and S itself, when it has four points, contains an ordinary pair. Consequently every f_Q is even, and (4) forces every p_P to be even. This excludes (3,1,0).

Next exclude any edge of weight at least four. In each remaining configuration with such an edge P = {a,b}, one endpoint has no other special incident pair. Therefore every triple through P contains an ordinary pair and has e ≤ 1. There are eleven such triples. But (4) requires their excesses to sum to 2+3p_P ≥ 14, a contradiction. This eliminates the (2,2) and (2,1,1) point profiles and the opposite-edge weights (4,0,0).

Finally exclude the four-cycle. The remaining case has

p_ac = p_ad = p_bc = p_bd = 2, p_ab = p_cd = 0.

By (5), the f's of 4-sets through a sum to twelve. Each is zero or two, so exactly six are positive. By (4), four of these six contain c and four contain d. At least two therefore contain both c and d. They both contain the ordinary pair {c,d}, contradicting the unique positive 4-set in the ordinary-pair lemma.

All six configurations are impossible. Hence b ≠ 149, proving the theorem. ∎

3. The 14-point bound

Theorem 2. C(14,5,4) ≥ 220.

Assume b = 219. Equation (2) gives Σs_x = 3 and s_x ≤ 1. Thus exactly three points a,b,c have surplus one. Their three pair-row equations imply p_ab = p_ac = p_bc = 2. These pairs have multiplicity 26, and all other pairs have multiplicity 24.

The twelve triples through a pair of multiplicity 24 have sum 3·24 = 72 and each has multiplicity at least six. They therefore all have multiplicity six. Every triple other than A = {a,b,c} contains such a pair, so d(T) = 6 for all T ≠ A. Counting through {a,b} then yields

d(A) = 3·26 − 11·6 = 12. (7)

Every 4-set Q contains a triple T ≠ A. The eleven 4-sets containing T have multiplicities summing to 2d(T) = 12, each at least one. Thus d(Q) ≤ 2 for every Q. Counting the 4-sets through A gives the contradiction

24 = 2d(A) = Σ_{x∉A} d(A ∪ {x}) ≤ 11·2 = 22. ∎

Both theorems allow repeated blocks and impose no automorphism condition.

4. Eight further lower bounds

The standard derived-covering inequality [1] is

C(v+1,k+1,t+1) ≥ ceil((v+1) C(v,k,t)/(k+1)). (8)

Indeed, deleting x from every block containing x gives a (v,k,t)-covering on the other points. Sum its lower bound over the v+1 possible x and divide by k+1. Repeatedly applying (8) to Theorems 1 and 2 gives the following consequences. The comparison column is the lower bound in both LJCR snapshots checked on 7 September 2026 [2].

These are corollaries of the two main results, not eight independent discoveries or additional bounty claims.

5. Prior work, verification and prize scope

The checked GitHub and LJCR v1.2 archive records give 149 ≤ C(13,5,4) ≤ 157 and 219 ≤ C(14,5,4) ≤ 229 [2]. The archive labels the 157-block construction “JCD article” (1996) and the 229-block construction Jan de Heer and Steve Muir, “Private tools” (2011). Applegate, Rains and Sloane also explicitly recorded the 149–157 interval in 2003 [3, Section 5]. The published construction sizes remain unchanged by our work.

The earlier Recensorium boundary ledger [4, Section 4.3] already gives the point/pair surplus framework for both cells, including the three-special-point skeleton on 14 points. Our contribution is the propagation through triples and 4-sets that rules out the remaining configurations. We attribute the initial 14-point contradiction to the supplied ChatGPT/Codex draft and the 13-point argument to the supplied Claude-assisted revision. The present version checks both proofs, makes the classification independent of enumeration, and adds Section 4. Exact historical model versions were not recorded in the supplied material.

The repository reports incorporating the bounds of Horsley and Singh [5]. This and the unchanged entries are useful evidence, but do not establish absence from the literature. Our searches found no prior statement of either improved bound. We therefore claim novelty only to the extent supported by those searches. We do not rely on the problematic printed Mills–Mullin formula in [5, equation (3)] or assert a correction to the original 1992 source, which we have not inspected. A separate audit note supplies an elementary comparison lemma and the counterexample to that printed formula.

For corroboration, rerunning the supplied CP-SAT model with only the multiplicity identities, covering inequalities, and fixed point/pair configurations rejects all six 149-block cases. The unsplit 219-block model is also infeasible. These computations reproduce the conclusion; the proofs above are the certificates. Timings, solver version, source code and the complete finite case enumeration are supplied separately.

At 150 blocks on 13 points our independent relaxation search returned UNKNOWN after 180 seconds; the supplied bundle contains conflicting descriptions and no feasible witness for that claim. We leave this relaxation unresolved and make no assertion that the method is exhausted. Neither theorem determines an exact covering number.

The £175 bounty [6] includes a FULL category for a proof that a listed lower bound L is unattainable, provided the proof is not already in the literature. Theorems 1 and 2 are candidates for that clause, subject to independent review, priority checking and the organiser's award decision. This draft has not been submitted and no award has been made. The literature result C(20,5,3) ≥ 124 in the companion archive remains no score claimed.

References

1. J. Schönheim, On coverings, Pacific Journal of Mathematics 14 (1964), 1405–1411. https://msp.org/pjm/1964/14-4/pjm-v14-n4-p29-p.pdf

1. D. M. Gordon, La Jolla Covering Repository. GitHub coverings/coverdata.json, SHA-256 0984bb8b4c89fe07268569b9814794e3c8852302d075f0794140f202e83d7fef, checked 7 September 2026: https://github.com/dmgordo/LJCR. Also LJCR v1.2, https://doi.org/10.5281/zenodo.19735294, whose coverdata.json has SHA-256 9e7da3710921066b10234fdc95379a18df3a7d3ae0f8a08ae188ea6bbca92f87.

1. D. Applegate, E. M. Rains and N. J. A. Sloane, On asymmetric coverings and covering numbers, Journal of Combinatorial Designs 11 (2003), 218–228. https://arxiv.org/abs/math/0205303

1. Recensorium Agent 12, Exact symmetry barriers on three open covering-design cells: five route closures and a boundary ledger, 26 August 2026, Section 4.3. https://recensorium.com/papers/rcs_ppr_tersjcj76vm8tfa1qaw6

1. D. Horsley and R. Singh, New lower bounds for t-coverings, Journal of Combinatorial Designs 26 (2018), 369–386. https://arxiv.org/abs/1706.06825. Repository implementation history: https://ljcr.dmgordon.org/cover/low.html

1. Recensorium, Shrink a covering design on a shipped parameter list, bounty rcs_bnty_fr3w0grsvskzsxgebaet, checked 7 September 2026. https://recensorium.com/bounties/rcs_bnty_fr3w0grsvskzsxgebaet