R(4,19) at n = 213: multiplier-invariant spaces for x ↦ 20x and x ↦ 11x exhausted

Result

The cell considered is R(4,19) at n = 213. The published lower bound in DS1 rev#18, dated 2026-04-24, in Radziszowski’s Small Ramsey Numbers, is R(4,19) ≥ 214, witnessed by a graph on 213 vertices. The computations reported here do not improve that bound.

The result is instead an exhaustive exclusion within two explicitly restricted spaces of Cayley graphs over Z_213. A candidate is (4,19)-free when it contains neither a K_4 nor an independent set of size 19. Either structure is called a violation.

For connection sets invariant under the multiplier map x ↦ 20x, the 106 inverse-pair representatives split into 11 multiplier orbits. The search space consisted of all 2047 non-empty unions of those orbits. All 2047 candidates were tested to completion across 3 independent shards. No candidate was (4,19)-free, and 0 candidates were unresolved. The best candidate had 140 violations.

For connection sets invariant under x ↦ 11x, the same 106 inverse-pair representatives split into 4 multiplier orbits. The search space consisted of all 15 non-empty unions of those orbits. All 15 candidates were tested to completion across 3 independent shards. Again, no candidate was (4,19)-free, and 0 candidates were unresolved. The best candidate had 54740 violations.

Thus neither specified multiplier-invariant space contains a circulant witness for R(4,19) ≥ 214.

Method

For a group G and an inverse-closed connection set S, the Cayley graph Cay(G,S) has vertex set G, with adjacency determined by differences lying in S. Here G is Z_213. Inverse pairs permit an undirected connection set to be represented by choices among 106 representatives.

A multiplier acts on these inverse-pair representatives. Requiring invariance under the multiplier means that a connection set cannot choose representatives independently: it must be a union of complete multiplier orbits. For x ↦ 20x, this condition reduces the enumeration to the 2047 non-empty unions of 11 orbits. For x ↦ 11x, it reduces the enumeration to the 15 non-empty unions of 4 orbits.

Cayley graphs are vertex-transitive. Consequently, testing whether Cay(G,S) contains K_s can be reduced to testing whether the neighbourhood of the identity contains K_{s-1}. In the present computation, this symmetry reduction makes the K_4 test substantially more direct: it suffices to seek the corresponding clique in the identity neighbourhood. Independent sets of size 19 were also tested as violations.

A candidate is conclusively rejected as soon as any violation is found. An unresolved candidate has a different and specific meaning: its violation count returned as zero even though its search did not complete. That situation did not occur. The multiplier-20 enumeration reached all 2047 candidates with 0 unresolved, and the multiplier-11 enumeration reached all 15 candidates with 0 unresolved. The negative conclusion therefore follows from exhaustive resolution of the stated candidate spaces, not from a search stopping before those spaces were covered.

The enumeration is deterministic. No language model, randomness, or heuristic search produced the reported figures. Re-running the same enumeration gives the same result.

Why this space

Multiplier-invariant circulants are a natural restricted class in computational Ramsey work. They combine vertex transitivity with additional algebraic symmetry, sharply organizing possible connection sets into orbit unions. This organization makes complete enumeration practical while retaining a class that includes important constructions from the published literature.

The Paley graphs illustrate the relevance of this restriction: their connection sets are exactly the quadratic residues and are invariant under multiplication by any square. Multiplier invariance is therefore not an arbitrary filtering device; it captures a familiar construction principle used in symmetric Ramsey graphs.

That motivation does not enlarge the formal scope of the result. It explains why these particular spaces were worth exhausting, not why success or failure within them should represent the unrestricted problem.

What this does not show

The multiplier-invariant spaces are a thin slice of the full connection-set space. An arbitrary inverse-closed connection set in Z_213 may select among the 106 inverse-pair representatives without being a union of the orbits induced by x ↦ 20x or x ↦ 11x. Such a connection set was not tested merely by virtue of this computation.

Accordingly, the result does not exclude all circulant graphs on 213 vertices. It excludes only the non-empty connection sets satisfying one of the two stated multiplier-invariance conditions. It says nothing about circulant connection sets outside those conditions.

Still less does the result exclude general graphs on 213 vertices. Cayley graphs over Z_213 are already a restricted family, and multiplier-invariant Cayley graphs are a further restricted family within it. Exhausting these spaces therefore cannot establish nonexistence of a general (4,19)-free graph on 213 vertices.

The violation totals should also be read only as outcomes within the enumerated classes. The best multiplier-20 candidate had 140 violations, while the best multiplier-11 candidate had 54740 violations. These values do not imply a bound for candidates outside the corresponding spaces.

The precise conclusion is limited but complete: every candidate in each declared non-empty orbit-union space was resolved, and none was (4,19)-free. No Ramsey bound was improved.

Reproduction

To reproduce the computation, begin with Z_213 and represent inverse-closed connection sets through the 106 inverse-pair representatives. For the first enumeration, apply x ↦ 20x to those representatives, accounting for inverse-pair identification, and recover the stated partition into 11 orbits. Enumerate all 2047 non-empty unions of these orbits. For each union, construct the associated Cayley graph and test for a K_4 or an independent set of size 19.

Repeat the procedure with x ↦ 11x. The induced partition has 4 orbits, yielding 15 non-empty orbit unions. Test every corresponding graph by the same violation criterion.

The work may be partitioned across 3 independent shards as in the reported runs. The complete multiplier-20 run must reach 2047 candidates, and the complete multiplier-11 run must reach 15 candidates. In each case, the unresolved count must be 0. The resulting searches find no (4,19)-free candidate; their best reported violation counts are respectively 140 and 54740.

Because the enumeration and tests are deterministic, reproduction does not require a random seed, a heuristic schedule, or model-generated choices. The essential reproducibility conditions are the group, the multiplier, the inverse-pair convention, exhaustive enumeration of every non-empty orbit union, and conclusive testing of every candidate.

Machine-readable coverage record

Appended verbatim by the harness, not written by the drafting model. Re-run the enumeration to check it.

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  "n": 213,
  "source_revision": "DS1 rev#18, 2026-04-24 (Radziszowski, Small Ramsey Numbers)",
  "improved": false,
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