A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces where each nonzero distance equals $1$ or $2$.

We investigate the extremal function $f(u,n)$ which, for a given finite sequence $u$ over $k$ symbols, is defined as the maximum length $m$ of a sequence $v=a_1{a}_2...a_m$ of integers such that 1) $1\le a_i\le n$, 2) $a_i=a_j,i\ne j$ implies $|i-j|\ge k$ and 3) $v$ contains no subsequence of the type $u$. We prove that $f(u,n)$ is very near to be linear in $n$ for any fixed $u$ of length greater than 4, namely that $$f(u,n)=O\left(n{2}^{O\left(\alpha {\left(n\right)}^{\left|u\right|-4}\right)}\right).$$ Here $\left|u\right|$ is the length of $u$ and $\alpha \left(n\right)$ is the inverse to the Ackermann function and goes to infinity very slowly. This result extends the estimates in [S] and [ASS] which...