Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.16803

The weak cylinder conjecture for polynomial graphs of bounded degree

Daniele Bartoli·Alessandro Giannoni
View original source ↗Read source document ↗Export metadata ↓

Abstract

Ball's weak cylinder conjecture asserts that a set of $p^2$ points in $\mathrm{AG}(3,p)$ with at least $p$ non-determined directions is a union of $p$ parallel lines. We prove the conjecture for sets admitting a polynomial graph representation of total degree at most $D$, provided that $p>D^{9/2}$. More precisely, a non-cylindrical graph of degree $D\ge3$ satisfies $|\mathcal{N}_S|\le p+1-\lceil p/(D-1)\rceil$ under the same hypothesis. This bound is attained when $p\equiv1\pmod{D-1}$. The proof relates non-determined directions to permutation polynomials on parallel affine lines. A rigidity lemma for moving line restrictions, together with a reconstruction of their compositional components, bounds the number of contributing directions uniformly in $p$. Directions with affine restrictions are treated separately; all other contributions are bounded using geometric indecomposability and the Weil bound for plane curves.

Explore related subjects

math.COmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Daniele Bartoli, Alessandro Giannoni. 2026-10-12. The weak cylinder conjecture for polynomial graphs of bounded degree. https://arxiv.org/abs/2610.16803

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Monochromatic non-commuting products

We show that a finite coloring of an amenable group contains `many' monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with more variables. This gives the first combinatorial proof and extensions of Bergelson and McCutcheon's non-commutative Schur theorem. Our main new tool is the introduction of what we call `quasirandom colorings,' a condition that is automatically satisfied by colorings of quasirandom groups, and a reduction to this case.

math.CO↗
arXiv2026

Complete tripartite subgraphs of balanced tripartite graphs with large minimum degree

In 1975 Bollobás, Erdős, and Szemerédi asked what minimum degree guarantees an octahedral subgraph $K_3(2)$ in any tripartite graph $G$ with $n$ vertices in each vertex class. We show that $δ(G)\geq n+2n^{\frac{5}{6}}$ suffices thus improving the bound $n+(1+o(1))n^{\frac{11}{12}}$ of Bhalkikar and Zhao obtained by following their approach. Bollobás, Erdős, and Szemerédi conjectured that $n+cn^{\frac{1}{2}}$ suffices and there are many $K_3(2)$-free tripartite graphs $G$ with $δ(G)\geq n+cn^{\frac{1}{2}}$. We confirm this conjecture under the additional assumption that every vertex in $G$ is adjacent to at least $(1/5+\varepsilon)n$ vertices in any other vertex class.

math.CO↗
arXiv2026

Almost-full transversals in equi-$n$-squares

In 1975, Stein made a wide generalisation of the Ryser-Brualdi-Stein conjecture on transversals in Latin squares, conjecturing that every equi-$n$-square (an $n\times n$ array filled with $n$ symbols where each symbol appears exactly $n$ times) has a transversal of size $n-1$. That is, it should have a collection of $n-1$ entries that share no row, column, or symbol. In 2017, Aharoni, Berger, Kotlar, and Ziv showed that equi-$n$-squares always have a transversal with size at least $2n/3$. In 2019, Pokrovskiy and Sudakov disproved Stein's conjecture by constructing equi-$n$-squares without a transversal of size $n-\frac{\log n}{42}$, but asked whether Stein's conjecture is approximately true. I.e., does an equi-$n$-square always have a transversal with size $(1-o(1))n$? We answer this question in the positive. More specifically, we improve both known bounds, showing that there exist equi-$n$-squares with no transversal of size $n-Ω(\sqrt{n})$ and that every equi-$n$-square contains $n-n^{1-Ω(1)}$ disjoint transversals of size $n-n^{1-Ω(1)}$. Furthermore, we give results for the corresponding problem in globally-bounded colourings of complete graphs, as part of the wider framework of coloured complete directed graphs.

math.CO↗