QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice, that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not definite, not necessarily unimodular, $n>2$. I want to show that for each $N>0$ there exists a 2-dimensional sublattice $\Lambda_0\subset \Lambda$, also not definite, such that all vectors $x\in \Lambda_0$ satisfy $|(x,x)| > N$.

This question is motivated by considering a K3 surface (or a hyperkahler manifold) with 2-dimensional Picard lattice. It is known that the squares of minimal rational curves are bounded, and we want to find a manifold with Picard number 2 having no minimal rational curves.