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Single-graph rational exponents in mixed multicolour Ramsey theory
Abstract
For every rational number γ>1, we construct a fixed bipartite graph Hγ such that rk(Hγ;KN)=Θγ,k((N/log N)γ) for every fixed k≥ 3. Here rk(H;KN)=r(H,…,H,KN), with k copies of H. The lower bound uses a symmetric random algebraic graph which avoids a fixed rooted power and has sufficiently strong local density for the Alon-Rodl method. The main algebraic step establishes a nonzero obstruction within the full symmetric coefficient space; the upper bound uses the rooted graphs from the proof of the rational exponents conjecture. We also obtain induced even-cycle asymptotics in the same construction.
Related
BibTeX
@misc{yang2026singlegraph,
title = {{Single-graph rational exponents in mixed multicolour Ramsey theory}},
author = {Yuang Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26061},
eprint = {2610.26061},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26061},
}
RIS
TY - UNPB
TI - Single-graph rational exponents in mixed multicolour Ramsey theory
AU - Yuang Yang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26061
UR - https://arxiv.org/abs/2610.26061
AB - For every rational number γ>1, we construct a fixed bipartite graph H_γ such that r_k(H_γ;K_N)=Θ_γ,k((N/log N)^γ) for every fixed k≥ 3. Here r_k(H;K_N)=r(H,…,H,K_N), with k copies of H. The lower bound uses a symmetric random algebraic graph which avoids a fixed rooted power and has sufficiently strong local density for the Alon-Rodl method. The main algebraic step establishes a nonzero obstruction within the full symmetric coefficient space; the upper bound uses the rooted graphs from the proof of the rational exponents conjecture. We also obtain induced even-cycle asymptotics in the same construction.
ER -