Single-graph rational exponents in mixed multicolour Ramsey theory

Abstract

For every rational number $\gamma>1$, we construct a fixed bipartite graph $H_\gamma$ such that $r_k(H_\gamma;K_N)=\Theta_{\gamma,k}((N/\log N)^\gamma)$ for every fixed $k\ge 3$. Here $r_k(H;K_N)=r(H,\ldots,H,K_N)$, 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.