We establish, for smooth projective real curves, an analogue of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.

Let ${ℳ}_g$ be the moduli space of smooth complex projective curves of genus $g$. Here we prove that the subset of ${ℳ}_g$ formed by all curves for which some Brill-Noether locus has dimension larger than the expected one has codimension at least two in ${ℳ}_g$. As an application we show that if $X\in {ℳ}_g$ is defined over $ℝ$, then there exists a low degree pencil $uX\to {\mathbb{P}}^1$ defined over $ℝ$.

It is consistent that ZF + DC holds, the hypergraph of rectangles on a given Euclidean space has countable chromatic number, while the hypergraph of equilateral triangles on ${ℝ}^2$ does not.