Przemek Chojecki | Searcharxiv

arXiv2026

The Second Term for Strongly 2-Primitive Sets

Let $F(n)$ be the largest size of a set $A\subseteq[1,n]$ such that $a\nmid bc$ whenever $a,b,c\in A$ and $a\notin\{b,c\}$, with $b$ and $c$ allowed to coincide. We prove \[ F(n)=\pi(n)+\left(\frac{27}{2}+o(1)\right)\frac{n^{2/3}}{(\log n)^2}. \] This determines the second-order constant conjectured by Erd\H{o}s; the upper bound keeps the leading constants in his multiplicative basis, while the lower bound packs scale-separated prime triples by proper edge-colourings.

math.NT↗

arXiv2026

Bene\v{s} and Shuffle-Exchange Counterexamples

We give explicit counterexamples to two rearrangeability conjectures for shuffle-type networks. First, for every $N\ge2$ we construct a simple $N$-regular ordered two-stage graph $L_N$ with $F(L_N)=2$ and $R(L_N)\ge N$, refuting the graph-theoretic Bene\v{s} inequality $R(L)\le2F(L)$ and its partition-stabilizer form as stated on Open Problem Garden. We retain the sharp cut obstruction, exact mask-composition identity, exact middle criterion, first nontrivial-level result, and balanced-middle sufficient condition that explain which extra hypotheses can replace mere external connectivity. Second, for the standard directed shuffle-exchange network, we prove $d(k,3)=6$ for every $k\ge3$, while the known binary value is $d(2,3)=5$. Hence the shuffle-exchange conjecture $d(k,n)=2n-1$ fails already at $(k,n)=(3,3)$, and the remaining upper bound $d(k,n)\le3n-3$ for $k\ge3$ suggests $d(k,n)=3n-3$ as a natural replacement problem.

math.CO↗