Given the r-distance graph on the hypercube \mathbb{F}_2^n, where two vertices are adjacent if their Hamming distance is exactly r, we study the maximum size T(n,r) of a triangle-free set of vertices. For even r\le n/2, we prove T(n,r)=O\!\left(\frac{r2^n}{n+1}\right).
We also obtain lower bounds in various regimes of r as a function of n.

We use p-rank bounds on partial ovoids and classical bounds on Ramsey numbers to obtain upper bounds on the size of partial m-ovoids in finite classical polar spaces. These bounds imply the non-existence of m-ovoids for new infinite families of polar spaces. We also give a probabilistic construction of large partial m-ovoids when m grows linearly with the rank of the polar space.

In the special case of symplectic spaces over the binary field, we prove an equivalence between partial m-ovoids and a generalisation of Oddtown families from extremal set theory, studied under the name of m-nearly orthogonal sets. We give a new construction for large partial 2-ovoids in these spaces, and thus for 2-nearly orthogonal sets over the binary field. This construction uses triangle-free graphs associated with certain BCH codes, whose complements have low 2-rank, and it gives an asymptotic improvement over the previous best construction.

We give another construction of triangle-free graphs using a binary projective cap, which has low complementary rank over the reals. This improves the bounds in the recently introduced rank-Ramsey problem and provides better constructions of large partial m-ovoids for m > 2 in the binary symplectic space.