Optimal Multivalued Shattering
  We have found the most general extension of the celebrated Sauer, Perles and Shelah, Vapnik and Chervonenkis result from 0-1 sequences to $k$-ary codes still giving a polynomial bound. Let $\mathcal{C}\subseteq \{0,1,..., k-1}^n$ be a $k$-ary code of length $n$. For a subset of coordinates $S\subset{1,2,...,n}$ the projection of $\mathcal{C}$ to $S$ is denoted by $\mathcal{C}|_S$. We say that $\mathcal{C}$ $(i,j)$-{\em shatters} $S$ if $\mathcal{C}|_S$ contains all the $2^{|S|}$ distinct vectors (codewords) with coordinates $i$ and $j$. Suppose that $\mathcal{C}$ does not $(i,j)$-shatter any coordinate set of size $s_{i,j}\geq 1$ for every $1\leq i< j\leq q$ and let $p=\sum (s_{i,j}-1)$. Using a natural induction we prove that $$ |{\mathcal C}|\leq O(n^p)$$ for any given $p$ as $n\to \infty$ and give a construction showing that this exponent is the best possible. Several open problems are mentioned. 
Zolt\'an F\"uredi (1 and 2) and Attila Sali (2) ((1) Department of   Mathematics University of Illinois at Urbana-Champaign,...
math.CO