Polynomial families of incident flags and explicit off-diagonal Ramsey graphs

@misc{verbeken2026polynomial,
  title = {{Polynomial families of incident flags and explicit off-diagonal Ramsey graphs}},
  author = {Brecht F. Verbeken},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06081},
  eprint = {2610.06081},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06081},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Polynomial families of incident flags and explicit off-diagonal Ramsey graphs
AU  - Brecht F. Verbeken
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06081
UR  - https://arxiv.org/abs/2610.06081
AB  - We construct explicit off-diagonal Ramsey graphs from polynomial families of incident point--hyperplane flags. Universal interpolation translates the exact dimension of ordered clique configurations into a bound on their coefficient--label incidence. At the critical dimension, a polynomial separates the forbidden-pair image from its diagonal; deleting the corresponding edges preserves a triangular rank certificate. Restriction of scalars realizes the rational endpoint without rounding loss. The resulting fixed-s exponent has leading scale s/(2log_2s), and short Frobenius relations give the explicit example R(16,t)≥Ω(t^2.0539221767…). A finite-fiber construction gives the range s^2(log s)^2=o(log t). For fixed parameters, algebraic preprocessing terminates and field initialization, vertex decoding and adjacency take deterministic time polynomial in the extension degree. We also retain an elimination-free variant and formulate the filtering argument for general forbidden configurations with a distinguished edge.
ER  -