arXiv:2602. 16807v2 Announce Type: replace Abstract: A collection of hyperplanes $\mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set $\{-1,1\}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $\mathcal{H}$ intersecting $e$ in its interior.
By Duncan Soiffer, Nathaniel Itty, Christopher D. Rosin, Blake Bruell, Mason DiCicco, G\'abor N. S\'ark\"ozy, Ryan Offstein, Daniel Reichman
arXiv:2608. 15558v1 Announce Type: cross Abstract: For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p.
By Zijian Zeng, Houde Liu, Kurunathan Ratnavelu
arXiv:2607. 21651v1 Announce Type: new Abstract: We prove that the maximum of $n$ real numbers is exactly representable by a ReLU network with two hidden layers for every $n\le 10$.
By Kilian Rue{\ss}, Gennadiy Averkov, Florestan Brunck, Moritz Grillo, Christoph Hertrich, Georg Loho, Jack Stade, Moritz Stargalla, Matthew Sun, Martin Winter
arXiv:2607. 22988v1 Announce Type: cross Abstract: In Problem~6 of his 1988 paper on differential posets, Stanley asked for the least possible cardinality of a fixed rank of an $r$-differential poset and suggested that the minimum should be attained by $Y^r$, the $r$-fold Cartesian power of Young's lattice.
By Xinan Dai, Yuchen Yang, Wenhao Deng, Yingdong Shi, Tailin Wu
arXiv:2608. 10869v1 Announce Type: new Abstract: Worst-case multiclass bounds do not become smaller when the best classifier is already nearly correct: what is missing is an optimistic rate, a guarantee whose fluctuation scales with the oracle risk itself.
By Xiaoyu Li, Andi Han, Jiaojiao Jiang, Junbin Gao
arXiv:2607. 21517v1 Announce Type: cross Abstract: The Shannon capacity $\Theta(G)$ of a graph $G$ quantifies the maximum rate at which information can be transmitted with zero error over a noisy channel.
By Nathaniel Itty, Christopher D. Rosin, Chase Carstensen, Daniel Reichman