arXiv subjects

Ricky Cipollini

Publications and source records attributed to Ricky Cipollini.

2 recordsLinked to original sources

arXiv2026

Optimality of Wouter van Doorn's Upper Bound for the Mayer-Erd\H{o}s Farey Problem

Let $\mathcal{F}_n$ be the Farey sequence of order $n$, written in increasing order. Call two fractions $\frac{a}{b} < \frac{c}{d}$ badly ordered if $a < c$ and $b > d$. Let $f(n)$ be the minimum number of Farey fractions strictly between two badly ordered fractions in $\mathcal{F}_n$. We prove $f(n)=\left(\frac{1}{4}+o(1)\right)n$. In the equivalent indexing convention of Erd\H{o}s Problem 1005, this determines the requested asymptotic constant as $c=1/4$. The upper bound $f(n)\le n/4+O(1)$ was first obtained by Wouter van Doorn; the main result here is the matching lower bound.

math.NT↗

arXiv2026

A sharp 5/8 bound for an Erd\H{o}s-S\'os pairwise-sums problem

Let $f_3(N)$ be the least integer such that every set $A\subseteq\{1,\ldots,N\}$ of size at least $f_3(N)$ contains distinct elements $a,b,c\in A$ such that $a+b\in A$, $a+c\in A$, and $b+c\in A$. We prove that $f_3(N)\le 5N/8+O(1)$. Together with the standard construction $[N/8,N/4]\cup[N/2,N]$, this gives $f_3(N)=5N/8+O(1)$, resolving Erd\H{o}s Problem 865. The proof is self-contained. An earlier conditional version of the reduction has also been formalized in Lean 4/Mathlib with no sorries and no added axioms.

math.CO↗