Kolja Knauer

Sara Asensio, Yuval Filmus, Ignacio García-Marco et al.

For any $m\geq 3$ we show that the Hamming graph $H(n,m)$ admits an imbalanced partition into $m$ sets, each inducing a subgraph of low maximum degree. This improves previous results by Tandya and by Potechin and Tsang, and disproves the Strong $m$-ary Sensitivity Conjecture of Asensio, García-Marco, and Knauer. On the other hand, we prove their weaker $m$-ary Sensitivity Conjecture by showing that the sensitivity of any $m$-ary function is bounded from below by a polynomial expression in its degree.

Kolja Knauer, Torsten Ueckerdt

A set $S\subseteq V$ of vertices of a graph $G$ is a \emph{$c$-clustered set} if it induces a subgraph with components of order at most $c$ each, and $α_c(G)$ denotes the size of a largest $c$-clustered set. For any graph $G$ on $n$ vertices and treewidth $k$, we show that $α_c(G) \geq \frac{c}{c+k+1}n$, which improves a result of Wood [arXiv:2208.10074, August 2022], while we construct $n$-vertex graphs $G$ of treewidth~$k$ with $α_c(G)\leq \frac{c}{c+k}n$. In the case $c\leq 2$ or $k=1$ we prove the better lower bound $α_c(G) \geq \frac{c}{c+k}n$, which settles a conjecture of Chappell and Pelsmajer [Electron.\ J.\ Comb., 2013] and is best-possible. Finally, in the case $c=3$ and $k=2$, we show $α_c(G) \geq \frac{5}{9}n$ and which is best-possible.

Victor Chepoi, Kolja Knauer, Manon Philibert

We show that the topes of a complex of oriented matroids (abbreviated COM) of VC-dimension $d$ admit a proper labeled sample compression scheme of size $d$. This considerably extends results of Moran and Warmuth on ample classes, of Ben-David and Litman on affine arrangements of hyperplanes, and of the authors on complexes of uniform oriented matroids, and is a step towards the sample compression conjecture -- one of the oldest open problems in computational learning theory. On the one hand, our approach exploits the rich combinatorial cell structure of COMs via oriented matroid theory. On the other hand, viewing tope graphs of COMs as partial cubes creates a fruitful link to metric graph theory.