The Burr-Erdős hypercube Ramsey conjecture

Theorem 1.1: there is an absolute constant $C$ with $R(Q_n)\le C2^n$ for every $n\ge0$; with the two-block lower bound $3\cdot2^{n-1}-1$ this determines the order of $R(Q_n)$. Consequences: linear Ramsey bounds for subgraphs occupying a fixed fraction of a cube, such as Cartesian powers of paths and other subgraphs of $Q_d$. Not shown: any numerical value of $C$, whether $R(Q_n)/2^n$ converges, or the general Burr-Erdős question for other dense families.

Theorem 1.1: there is an absolute constant $C$ with $R(Q_n)\le C2^n$ for every $n\ge0$; with the two-block lower bound $3\cdot2^{n-1}-1$ this determines the order of $R(Q_n)$. Consequences: linear Ramsey bounds for subgraphs occupying a fixed fraction of a cube, such as Cartesian powers of paths and other subgraphs of $Q_d$. Not shown: any numerical value of $C$, whether $R(Q_n)/2^n$ converges, or the general Burr-Erdős question for other dense families.