Metric betweenness on five points: a complete census graded by collinear triples
Abstract
A ternary relation B on a set X is a metric betweenness relation if there is a metric d on X with B(x,z,y) ⇔ d(x,z)+d(z,y)=d(x,y). Vlasic (arXiv:allowbreak()2607.27222) enumerated these relations on finite sets by linear programming, obtaining bₙ = 1, 1, 4, 74, 8628, … for n ≤ 6, and quoted five axioms (B0)–(B4) of Mendris and Zlatov s that every metric betweenness relation satisfies. It is known that the axioms are not sufficient, that the smallest failures occur at six points (Mendris–Zlatov s), and that at five points they are exactly right (Chvátal, who counted 122 isomorphism types). We refine the five-point count into a complete census graded by the number k of collinear triples, that is, of triples one of whose points lies between the other two: for k = 0, 1, …, 10 the numbers of relations on a five-point set satisfying (B0)–(B4) are 1, 30, 285, 1100, 2010, 2142, 1640, 840, 390, 130, 60, summing to 8628. The same eleven numbers count the metric betweenness relations on five points, because we also prove — by exhibiting, for each of the 8628 structures, an explicit integer metric on {0,1,2,3,4} with all ten distances at most 7 — that at five points "satisfies (B0)–(B4)" and "is a metric betweenness relation" are the same condition. Every statement of this kind below is machine-checked in Lean 4, including the proof that the encoding used by the search is a bijection onto the class it is meant to count, and controls showing that neither (B3) nor (B4) is redundant. Separately, and outside the formal development, we record six-point data: b₆ = 7221418, which agrees with the corrected entry A395237 of the OEIS and not with the value 7238428 printed in the source; the number 8896888 of six-point relations satisfying (B0)–(B4), of which 1675470 are not metrizable; and the six-point distribution by k, whose zero at k = 19 is the case n = 6 of a theorem of Szabó.
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Source snapshot 2026-09-07 03:53 UTC
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Claim ledger
Stated results
MB1known2026-09-03
The five Mendris-Zlatos betweenness axioms (B0)-(B4), and uniqueness of the middle, hold for every metric-induced betweenness relation on an arbitrary set
MB2routine2026-09-03
Additivity of the betweenness defect: B_(d1+d2) = B_(d1) cap B_(d2), so the metrizable betweenness relations on any set form a closure system, and are invariant under positive scaling
MB3routine2026-09-03
Witnesses and heredity: injective real coordinates give a metric whose betweenness is the order on the line, the discrete metric realises the trivial relation, and metrizability restricts to subsets
MB4routine2026-09-03
Negative controls: neither the all-true nor the all-false ternary relation is metrizable, and metrizability is NOT preserved by union
MB5known2026-09-03
A six-point relation satisfying the Mendris-Zlatos axioms (B0)-(B4) that no metric realises, machine-checked, with all six of its five-point restrictions metrizable by explicit integer metrics
MB6routine2026-09-03
That witness is minimal under deleting a betweenness triple: dropping any one of its four triples yields a metrizable relation, by four explicit integer metrics on six points
MB7correction2026-09-03
b₆ = 7221418: the number of metrizable betweenness relations on six points, recomputed independently; the value 7238428 printed in arXiv:2607.27222v1 Table 1 is wrong
This ledger entry is reported in prose and is not bound to a Lean theorem.MB8known data2026-09-03
Independent confirmation of A397184(6) = 11610, A395250(6) = 6460153, A395485(6) = 10287, of the extreme-ray counts 3, 7, 25, 296 of METₙ, and of the corrected orbit count 8 for MET₆
This ledger entry is reported in prose and is not bound to a Lean theorem.MB9candidate2026-09-03
bₙ counts the full-support faces of the metric cone METₙ; total face counts f(METₙ) = 8, 106, 9484, 7356040 for n = 3..6, and the identity sumₖ S(n,k) bₖ = f(METₙ)
This ledger entry is reported in prose and is not bound to a Lean theorem.MB10candidate2026-09-03
The f-vectors of MET₃, MET₄, MET₅, MET₆ by dimension, and their full-support parts (bₙ graded by face dimension)
This ledger entry is reported in prose and is not bound to a Lean theorem.MB11candidate2026-09-03
The distribution of metrizable betweenness relations on n points by the number k of triples that have a middle, for n = 4, 5, 6, labelled and up to isomorphism; and the gap b_(6,19) = 0
This ledger entry is reported in prose and is not bound to a Lean theorem.MB12candidate2026-09-03
8896888 relations on six points satisfy the Mendris-Zlatos axioms (B0)-(B4), of which 1675470 are not metrizable; the 270 with the fewest betweenness triples are exactly Chvatal's two obstructions B1 and B2
This ledger entry is reported in prose and is not bound to a Lean theorem.MB13measurement2026-09-03
Cost: the face-lattice route computes all four n = 6 sequence values, the S₆ orbits, the convexities and the f-vector in 127 CPU-seconds and 415 MB
This ledger entry is reported in prose and is not bound to a Lean theorem.MB14routine2026-09-03
Every pseudometric betweenness relation splits into its zero-kernel (an equivalence relation, readable off the relation itself) and a genuine metrizable relation on class representatives – the combinatorial core of sumₖ S(n,k) bₖ = f(METₙ)
MB15known data2026-09-03
Exactly 8628 ternary relations on a five-point set satisfy the Mendris-Zlatos axioms (B0)-(B4), machine-checked by exhaustion over the 4¹0 slot codes, with the code/relation bijection proved and neither (B3) nor (B4) redundant
MB16candidate2026-09-03
The k-graded census at n = 5 in the kernel: the number of ternary relations on a five-point set satisfying (B0)-(B4) with exactly k collinear triples is 1, 30, 285, 1100, 2010, 2142, 1640, 840, 390, 130, 60 for k = 0..10
MB17known2026-09-03
Every ternary relation on a five-point set satisfying (B0)-(B4) is the betweenness relation of a metric – 8628 explicit integer metrics on 0,1,2,3,4 with entries at most 7, machine-checked – so at five points metrizable and axiom-satisfying coincide, and b₅ = 8628 is a count of metrics
MB18candidate2026-09-03
The k-graded census of METRIZABLE betweenness relations on five points, in the kernel: for k = 0..10 the number of betweenness relations of a metric on a five-point set with exactly k collinear triples is 1, 30, 285, 1100, 2010, 2142, 1640, 840, 390, 130, 60
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Counting the ternary relations on a finite set that are betweenness relations of a metric, and the polyhedral structure that makes the count exact.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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