Triangle-free Rank--Ramsey graphs from ordered incidence lifts

Abstract

For a graph $G$ with adjacency matrix $A_G$ and a coefficient field $\mathbb K$, put $ρ_{\mathbb K}(G)=\operatorname{rank}_{\mathbb K}(A_G+I)$. We answer the triangle-free Rank--Ramsey existence question of Beniamini, Linial and Shraibman. For every order $n$, we construct a triangle-free graph $H_n$ such that, simultaneously over all coefficient fields, $$ ρ_{\mathbb K}(H_n)=\bigl(4(2/3)^{3/4}+o(1)\bigr)n^{3/4}, $$ with an error uniform in the field. The graphs can be chosen Hamiltonian for all sufficiently large $n$. Our method combines ordered layers of point--line incidence flags with a common low-dimensional endpoint space. We determine the exact real rank, independence number and chromatic number of the affine lifts. For all triangle-free complete incidence lifts, we obtain a sharp two-term lower bound for complete incidence lifts, with sharpness witnessed by projective-plane incidence graphs over suitable coefficient fields. We also establish a third-order estimate and stability near equality. On the same explicit graphs, both the Lovász number and minimum semidefinite rank exceed the prescribed Boolean-matrix rank by polynomial factors. Thus neither parameter has a universal constant-factor upper bound in terms of complement rank. We determine Shannon capacity in the short-layer range.