Stijn Cambie

Avoiding a partial Latin square with an isotope of a given Latin square

Stijn Cambie, Ian M. Wanless

For all sufficiently large $n$ , we prove that every Latin square of order $n$ can be made to avoid any prescribed partial Latin square of order $n$ by permuting only its rows and c…

math.HO2026

Short Proofs for the Minimum Number of Determined Circles

Stijn Cambie, Denis Paliy, Rafał Wrona

We present short proofs for the main parts of the recently completed solution of Erdős Problem 506, which asks for the minimum number of distinct circles determined by a finite pla…

Four collinear points or six visible points

Stijn Cambie

Every finite planar set of at least $880$ points contains four collinear points or six pairwise visible points.

Brown-Gerver-Ramsey Theorems in Small Dimensions

Stijn Cambie, Erik Kalviainen, J. Shallit

We consider infinite walks in $N^{k}$ with standard unit basis vector steps that avoid $t$ collinear points, and show that these walks exist for $(k,t) \in \{(6,3), (4,4), (…

An infinite small-step $Z^{3}$ -walk with no collinear triple

Stijn Cambie, Erik Kalviainen

We construct an infinite walk in $Z^{3}$ whose steps come from a fixed set of sixteen vectors and no three of whose vertices are collinear, answering a problem of Gerver and…

The number and average length of subpaths in graphs

Stijn Cambie

We study extremal questions on the average length and the number of subpaths in a graph. In particular, we prove the questions of Jamison (from $1983, 1984$ ) for the analogous conc…

math.NT2025

Resolution of two conjectures by Erdős and Hall concerning separable numbers

Stijn Cambie, Wouter van Doorn

Erdős and Hall defined a pair $(m, n)$ of positive integers to be interlocking, if between any pair of consecutive divisors (both larger than $1$ ) of $n$ (resp. $m$ ) there is a div…

On the extrema of the mean subtree order of graphs

Stijn Cambie, Jorik Jooken, Stephan Wagner

It has been conjectured that the minimum and maximum of the mean subtree order among connected graphs of order $n$ are attained by the path $P_{n}$ and clique $K_{n}$ , respectively. Ex…

Proving it is impossible; on Erdős problem $#278$

Stijn Cambie

Erdős and Graham asked for the minimum density missed by one chosen residue class for each of a prescribed collection of moduli. We give exact expressions for natural structured fa…

Note on extremal problems about connected subgraph sums

Stijn Cambie, Carla Groenland

For a graph $G$ with vertex assignment $c : V (G) \to Z^{+}$ , we define $\sum_{v \in V (H)} c (v)$ for $H$ a connected subgraph of $G$ as a connected subgraph sum of $G$ . We study t…

Uniform Šoltés' hypergraphs and Šoltés' weighted graphs

Stijn Cambie, Ajay Tiwari

A Šoltés' hypergraph is a hypergraph for which the removal of any of its vertices does not change its total distance. We prove that every uniform Šoltés' hypergraph has order at le…

Reconfiguration of List Colourings

Stijn Cambie, Wouter Cames van Batenburg, Daniel W. Cranston +2

Given a proper (list) colouring of a graph $G$ , a recolouring step changes the colour at a single vertex to another colour (in its list) that is currently unused on its neighbours,…

On edge-colouring-games by Erdős, and Bensmail and Mc Inerney

Stijn Cambie, Michiel Provoost

We study two games proposed by Erdős, and one game by Bensmail and Mc Inerney, all sharing a common setup: two players alternately colour edges of a complete graph, or in the biase…

On the order of intersecting hypergraphs

Stijn Cambie, Jaehoon Kim, Hyunwoo Lee +2

Determining the maximum number of edges in an intersecting hypergraph on a fixed ground set under additional constraints is one of the central topics in extremal combinatorics. In…

math.NT2025

On Erdős problem $#648$

Stijn Cambie

We establish the order of the maximum length of an increasing sequence, bounded by $n$ , in which the largest prime divisor of the elements form a decreasing sequence.

Sharp results for the Erdős, Pach, Pollack and Tuza problem

Stijn Cambie, Jorik Jooken

We consider the Erdős, Pach, Pollack and Tuza problem, asking for the maximum diameter of a graph with given order $n$ , minimum degree $δ$ and clique number at most $ω$ . We solve t…

math.NT2025

Resolution of Erdős' problems about unimodularity

Stijn Cambie

Letting $δ_{1} (n, m)$ be the density of the set of integers with exactly one divisor in $(n, m)$ , Erdős wondered if $δ_{1} (n, m)$ is unimodular for fixed $n$ . We prove this is false in ge…

Abundancy of $z$ -\v Soltés' digraphs

Stijn Cambie

We prove the existence of infinitely many \v Soltés' digraphs, the digraph analogue of \v Soltés' graphs. We also give an example of a \v Soltés' digraph with trivial automorphism…

Sharp Bounds on Lengths of Linear Recolouring Sequences

Stijn Cambie, Wouter Cames van Batenburg, Daniel W. Cranston

A recolouring sequence, between $k$ -colourings $α$ and $β$ of a graph $G$ , transforms $α$ into $β$ by recolouring one vertex at a time, such that after each recolouring step we aga…

Partitions of planar (oriented) graphs into a connected acyclic and an independent set

Stijn Cambie, François Dross, Kolja Knauer +2

A question at the intersection of Barnette's Hamiltonicity and Neumann-Lara's dicoloring conjecture is: Can every Eulerian oriented planar graph be vertex-partitioned into two acyc…

Open problems of the 32nd Workshop on Cycles and Colourings

János Barát, Stijn Cambie, Geňa Hahn +4

Since its beginnings, every Cycles and Colourings workshop holds one or two open problem sessions; this document contains the problems (together with notes regarding the current st…

math.GM2024

Resolution of an Erdős' problem on least common multiples

Stijn Cambie

Erdős posed the question whether there exist infinitely many sets of consecutive numbers whose least common multiple (lcm) exceeds the lcm of another, larger set with greater conse…

Fractional list packing for layered graphs

Stijn Cambie, Wouter Cames van Batenburg

The fractional list packing number $χ_{ℓ}^{∙} (G)$ of a graph $G$ is a graph invariant that has recently arisen from the study of disjoint list-colourings. It measures how…