George F. McNulty

A k-flat in a vector space is a k-dimensional affine subspace. Our basic result is that an injection \(T:{{\mathbb {C}}}^n\rightarrow {{\mathbb {C}}}^n\) that for some \(k\in \{1,2,\ldots ,n-1\}\), T maps all k-flats to flats of \({{\mathbb {C}}}^n\) and is either continuous at a point or Lebesgue measurable, is either an affine map or a conjugate-affine map. An analogous result is proven for injections of the complex projective spaces. In the case of continuity at a point, this is generalized in several directions, the main one being that the complex numbers can be replaced by a finite-dimensional division algebra over an Archimedean ordered field. We also prove injective versions of the Fundamental Theorems of affine and projective geometry and give a counter-example to the surjective version of the latter. This extends work of A. G. Gorinov on a problem of V. I. Arnold.

The set of all doubled patterns on $n$ or fewer letters can be avoided on an alphabet with $k$ letters, where $k$ is the least even integer strictly greater than $n+1$, with the exception of $n=4$. The set of all doubled patterns on $4$ or fewer letters can be avoided on the $8$-letter alphabet. The set of all avoidable patterns on $n$ or fewer letters can be avoided on an alphabet with $2(n+2)$ letters.