Improved Explicit and Algorithmic Lower Bounds for r(5,t)

Abstract

We modify a construction of Brada\v{c} \cite{Bradac26} using a vertex ordering to give, for every prime power $q$, an explicit $K_5$ free graph with more than $q^7$ vertices and independence number below $600q^4$. This shows $$r(5,t)=\Omega(t^{7/4}).$$ Additionally, pruning a graph defined by the Coulter and Matthews polynomial $X^{14}$ \cite{CM}, we give a deterministic polynomial time construction proving $$r(5,t)=\Omega\bigl(t^{20/11-\varepsilon}\bigr)$$ for every fixed $\varepsilon>0$. These results improve the constructive lower bound $\Omega(t^{5/3})$ of Kostochka, Pudl\'ak and R\"odl \cite{KPR}.