On the Tur\'an number of the hypercube

In 1964, Erd\H{o}s proposed the problem of estimating the Tur\'an number of the $d$-dimensional hypercube $Q_d$. Since $Q_d$ is a bipartite graph with maximum degree $d$, it follows from results of F\"uredi and Alon, Krivelevich, Sudakov that $\mathrm{ex}(n,Q_d)=O_d(n^{2-1/d})$. A recent general result of Sudakov and Tomon implies the slightly stronger bound $\mathrm{ex}(n,Q_d)=o(n^{2-1/d})$. We obtain the first power-improvement for this old problem by showing that $\mathrm{ex}(n,Q_d)=O_d(n^{2-\frac{1}{d-1}+\frac{1}{(d-1)2^{d-1}}})$. This answers a question of Liu. Moreover, our techniques give a power improvement for a larger class of graphs than cubes. We use a similar method to prove that any $n$-vertex, properly edge-coloured graph without a rainbow cycle has at most $O(n(\log n)^2)$ edges, improving the previous best bound of $n(\log n)^{2+o(1)}$ by Tomon. Furthermore, we show that any properly edge-coloured $n$-vertex graph with $\omega(n\log n)$ edges contains a cycle which is almost rainbow: that is, almost all edges in it have a unique colour. This latter result is tight.