Minimum degree five in a hypothetical (3,10)-graph of order 40

R(3,10) is 40 or 41, and which one turns on a single question: does a (3,10)-graph on 40 vertices exist? We contribute two things to that question.

A structural exclusion, unconditional. No (3,10)-graph on 40 vertices has a vertex of degree 4, so every such graph has minimum degree at least 5. Pandey and Ravi obtain the same exclusion under the hypothesis that the graph has diameter 2; we remove that hypothesis. The proof fixes the unique (3,9)-graph of order 35 as the dual neighbourhood, leaving 140 undetermined adjacencies, and refutes the resulting 98,965-clause instance in ten seconds. The refutation is machine-checked, the instance is independently re-derived from the definitions by a second implementation, and the whole chain reproduces cold in fourteen seconds on one core.

A specification of what would finish the job. The case analysis is exhaustive in six cases and we close one, which is not progress toward a Ramsey value and is not offered as such. What is offered is the audit: each remaining case reduces to a census of (3,9)-graphs over a window of edge counts — 79 (order, edge-count) entries in all. Eleven have been published as exact counts. None is in circulation as isomorphism-class representatives, and since the argument fixes a core and refutes, a count does not substitute for the graphs. We identify every entry, distinguish the eleven that were enumerated and not deposited from the sixty-eight that nobody has enumerated, and show that obtaining all eleven would still close no case. Anyone holding those censuses can finish a case with the computation given here; nobody can finish one without them.

MSC 2020: 05C55, 05C85, 68R10, 68V05.