arXiv:2607. 23004v1 Announce Type: cross Abstract: Let $t(N)$ be the largest $t$ for which there exist distinct sets $A_1,\dots,A_t \subseteq \{1,\dots,N\}$ such that $A_i \cap A_j$ is a nonempty arithmetic progression for all $i \neq j$ (Erdos Problem #272).
By Zhanfu Yang
arXiv:2608. 14369v1 Announce Type: cross Abstract: We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure $p$-spin glasses with overlap $q$.
By Taegyun Kim
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: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. 11211v1 Announce Type: new Abstract: Conway's 99-graph problem asks whether a strongly regular graph with parameters $\mathrm{srg}(99,14,1,2)$ exists.
By Aalok Thakkar
arXiv:2606. 18236v1 Announce Type: new Abstract: In learning theory, the sign rank of a binary concept class captures the smallest dimension in which it can be represented by points and halfspaces.
By Ari Blondal, Hamed Hatami, Pooya Hatami, Chavdar Lalov, Sivan Tretiak