In 1964, Erdős proposed the problem of estimating the Turán number of the d-dimensional hypercube
$Q_d$. Since
$Q_d$ is a bipartite graph with maximum degree d, it follows from results of Füredi 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
$\mathrm {ex}(n,Q_d)=O_d\left (n^{2-\frac {1}{d-1}+\frac {1}{(d-1)2^{d-1}}}\right )$. 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.