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We propose the following problem. For some k ≥ 1, a graph G is to be properly edge coloured such that any two adjacent vertices share at most k colours. We call this the k-intersection edge colouring. The minimum number of colours sufficient to guarantee such a colouring is the k-intersection chromatic index and is denoted χ’ₖ(G). Let fₖ be defined by $f ₖ (Δ) = m a x_{G : Δ (G) = Δ} χ^{'} ₖ (G)$ . We show that fₖ(Δ) = Θ(Δ²/k). We also discuss some open problems.

On monochromatic sets of integers whose diameters form a monotone sequence.

Bialostocki, Arie, Wilson, Bryan (2009)

Integers

On monochromatic solutions of equations in groups.

P. Cameron, J. Cilleruelo, O. Serra (2007)

Revista Matemática Iberoamericana

On Monochromatic Subgraphs of Edge-Colored Complete Graphs

Eric Andrews, Futaba Fujie, Kyle Kolasinski, Chira Lumduanhom, Adam Yusko (2014)

Discussiones Mathematicae Graph Theory

In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the mono- chromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that every red-blue coloring of Kn results in a monochromatic G or a monochromatic H. Thus, the standard Ramsey...

On non-intersecting arithmetic progressions

Régis de la Bretèche, Kevin Ford, Joseph Vandehey (2013)

Acta Arithmetica

We improve known bounds for the maximum number of pairwise disjoint arithmetic progressions using distinct moduli less than x. We close the gap between upper and lower bounds even further under the assumption of a conjecture from combinatorics about Δ-systems (also known as sunflowers).

On path-quasar Ramsey numbers

Binlong Li, Bo Ning (2015)

Annales UMCS, Mathematica

Let G1 and G2 be two given graphs. The Ramsey number R(G1,G2) is the least integer r such that for every graph G on r vertices, either G contains a G1 or Ḡ contains a G2. Parsons gave a recursive formula to determine the values of R(Pn,K1,m), where Pn is a path on n vertices and K1,m is a star on m+1 vertices. In this note, we study the Ramsey numbers R(Pn,K1,m), where Pn is a linear forest on m vertices. We determine the exact values of R(Pn,K1∨Fm) for the cases m ≤ n and m ≥ 2n, and for the case...

On rainbow arithmetic progressions.

Axenovich, Maria, Fon-Der-Flaass, Dmitri (2004)

The Electronic Journal of Combinatorics [electronic only]

On rainbow solutions to an equation with a quadratic term.

Zhan, Tong (2009)

Integers

On Ramsey $(K_{1, 2}, C ₄)$ -minimal graphs

Tomás Vetrík, Lyra Yulianti, Edy Tri Baskoro (2010)

Discussiones Mathematicae Graph Theory

For graphs F, G and H, we write F → (G,H) to mean that any red-blue coloring of the edges of F contains a red copy of G or a blue copy of H. The graph F is Ramsey (G,H)-minimal if F → (G,H) but F* ↛ (G,H) for any proper subgraph F* ⊂ F. We present an infinite family of Ramsey $(K_{1, 2}, C ₄)$ -minimal graphs of any diameter ≥ 4.

On randomly generated intersecting hypergraphs.

Bohman, Tom, Cooper, Colin, Frieze, Alan, Martin, Ryan, Ruszinkó, Miklós (2003)

The Electronic Journal of Combinatorics [electronic only]

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