A. Nicholas Day
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View article: Upper density of monochromatic paths in edge-coloured infinite complete graphs and bipartite graphs
Upper density of monochromatic paths in edge-coloured infinite complete graphs and bipartite graphs Open
The upper density of an infinite graph $G$ with $V(G) \subseteq \mathbb{N}$ is defined as $\overline{d}(G) = \limsup_{n \rightarrow \infty}{|V(G) \cap \{1,\ldots,n\}|}/{n}$. Let $K_{\mathbb{N}}$ be the infinite complete graph with vertex s…
View article: On a Conjecture of Nagy on Extremal Densities
On a Conjecture of Nagy on Extremal Densities Open
We disprove a conjecture of Nagy on the maximum number of copies N(G,H) of a fixed graph G in a large graph H with prescribed edge density. Nagy conjectured that for all G, the quantity N(G,H) is asymptotically maximised by either a quasi-…
View article: Long paths and connectivity in 1‐independent random graphs
Long paths and connectivity in 1‐independent random graphs Open
A probability measure on the subsets of the edge set of a graph G is a 1 ‐independent probability measure (1‐ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1‐ipm , deno…
View article: Maker–Breaker percolation games I: crossing grids
Maker–Breaker percolation games I: crossing grids Open
Motivated by problems in percolation theory, we study the following two-player positional game. Let Λ m × n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alte…