Katherine Staden
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View article: Ringel’s tree packing conjecture in quasirandom graphs
Ringel’s tree packing conjecture in quasirandom graphs Open
We prove that any quasirandom graph with n vertices and rn edges can be decomposed into n copies of any fixed tree with r edges. The case of decomposing a complete graph establishes a conjecture of Ringel from 1963.
View article: A framework for the generalised Erdős-Rothschild problem and a resolution of the dichromatic triangle case
A framework for the generalised Erdős-Rothschild problem and a resolution of the dichromatic triangle case Open
The Erdős-Rothschild problem from 1974 asks for the maximum number of $s$-edge colourings in an $n$-vertex graph which avoid a monochromatic copy of $K_k$, given positive integers $n,s,k$. In this paper, we systematically study the general…
View article: Universality for transversal Hamilton cycles
Universality for transversal Hamilton cycles Open
Let be a graph collection on a common vertex set of size such that for every . We show that contains every Hamilton cycle pattern. That is, for every map there is a Hamilton cycle whose th edge lies in .
View article: Stability of transversal Hamilton cycles and paths
Stability of transversal Hamilton cycles and paths Open
Given graphs $G_1,\ldots,G_s$ all on a common vertex set and a graph $H$ with $e(H) = s$, a copy of $H$ is \emph{transversal} or \emph{rainbow} if it contains one edge from each $G_i$. We establish a stability result for transversal Hamilt…
View article: Exact solutions to the Erdős-Rothschild problem
Exact solutions to the Erdős-Rothschild problem Open
Let $\boldsymbol {k} := (k_1,\ldots ,k_s)$ be a sequence of natural numbers. For a graph G , let $F(G;\boldsymbol {k})$ denote the number of colourings of the edges of G with colours $1,\dots ,s$ such that, for every $c \in \{1,\dots ,s\}$…
View article: Universality for transversal Hamilton cycles
Universality for transversal Hamilton cycles Open
Let $\mathbf{G}=\{G_1, \ldots, G_m\}$ be a graph collection on a common vertex set $V$ of size $n$ such that $δ(G_i) \geq (1+o(1))n/2$ for every $i \in [m]$. We show that $\mathbf{G}$ contains every Hamilton cycle pattern. That is, for eve…
View article: Stability from graph symmetrisation arguments with applications to inducibility
Stability from graph symmetrisation arguments with applications to inducibility Open
We present a sufficient condition for the stability property of extremal graph problems that can be solved via Zykov's symmetrisation. Our criterion is stated in terms of an analytic limit version of the problem. We show that, for example,…
View article: Transversals via regularity
Transversals via regularity Open
Given graphs $G_1,\ldots,G_s$ all on the same vertex set and a graph $H$ with $e(H) \leq s$, a copy of $H$ is transversal or rainbow if it contains at most one edge from each $G_c$. When $s=e(H)$, such a copy contains exactly one edge from…
View article: Stability for the Erdős-Rothschild problem
Stability for the Erdős-Rothschild problem Open
Given a sequence $\boldsymbol {k} := (k_1,\ldots ,k_s)$ of natural numbers and a graph G , let $F(G;\boldsymbol {k})$ denote the number of colourings of the edges of G with colours $1,\dots ,s$ , such that, for every $c \in \{1,\dots ,s\}$…
View article: Exact solutions to the Erdős-Rothschild problem
Exact solutions to the Erdős-Rothschild problem Open
Let $\textbf{k} := (k_1,\ldots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\textbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edge…
View article: Exact solutions to the Erd\H{o}s-Rothschild problem
Exact solutions to the Erd\H{o}s-Rothschild problem Open
Let $\textbf{k} := (k_1,\ldots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\textbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edge…
View article: Stability for the Erd\H{o}s-Rothschild problem
Stability for the Erd\H{o}s-Rothschild problem Open
Given a sequence $\\mathbf{k} := (k_1,\\ldots,k_s)$ of natural numbers and a\ngraph $G$, let $F(G;\\mathbf{k})$ denote the number of colourings of the edges\nof $G$ with colours $1,\\dots,s$ such that, for every $c \\in \\{1,\\dots,s\\}$, …
View article: Stability for the Erdős-Rothschild problem
Stability for the Erdős-Rothschild problem Open
Given a sequence $\mathbf{k} := (k_1,\ldots,k_s)$ of natural numbers and a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges …
View article: On the Maximum Number of Integer Colourings with Forbidden Monochromatic Sums
On the Maximum Number of Integer Colourings with Forbidden Monochromatic Sums Open
Let $f(n,r)$ denote the maximum number of colourings of $A \subseteq \lbrace 1,\ldots,n\rbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\lbrace x,y,z\rbrace$ such that $x+y=z$. We show that $f(n,2…
View article: Geometric constructions for Ramsey-Turán theory
Geometric constructions for Ramsey-Turán theory Open
Combining two classical notions in extremal combinatorics, the study of Ramsey-Turán theory seeks to determine, for integers $m\le n$ and $p \leq q$, the number $\mathsf{RT}_p(n,K_q,m)$, which is the maximum size of an $n$-vertex $K_q$-fre…
View article: Geometric constructions for Ramsey-Tur\'an theory
Geometric constructions for Ramsey-Tur\'an theory Open
Combining two classical notions in extremal combinatorics, the study of Ramsey-Turan theory seeks to determine, for integers $m\le n$ and $p \leq q$, the number $\mathsf{RT}_p(n,K_q,m)$, which is the maximum size of an $n$-vertex $K_q$-fre…
View article: Stability from graph symmetrisation arguments with applications to inducibility
Stability from graph symmetrisation arguments with applications to inducibility Open
We present a sufficient condition for the stability property of extremal graph problems that can be solved via Zykov's symmetrisation. Our criterion is stated in terms of an analytic limit version of the problem. We show that, for example,…
View article: Ringel's tree packing conjecture in quasirandom graphs
Ringel's tree packing conjecture in quasirandom graphs Open
We prove that any quasirandom graph with $n$ vertices and $rn$ edges can be decomposed into $n$ copies of any fixed tree with $r$ edges. The case of decomposing a complete graph establishes a conjecture of Ringel from 1963.
View article: The bandwidth theorem for locally dense graphs
The bandwidth theorem for locally dense graphs Open
The bandwidth theorem of Böttcher, Schacht, and Taraz [ Proof of the bandwidth conjecture of Bollobás and Komlós, Mathematische Annalen, 2009 ] gives a condition on the minimum degree of an n -vertex graph G that ensures G contains every r…
View article: Minimum Number of Additive Tuples in Groups of Prime Order
Minimum Number of Additive Tuples in Groups of Prime Order Open
For a prime number $p$ and a sequence of integers $a_0,\dots,a_k\in \{0,1,\dots,p\}$, let $s(a_0,\dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,\dots,x_k)\in A_0\times\dots\times A_k$ with $x_0=x_1+\dots + x_k$, over subsets $A_…
View article: Independent Sets in Hypergraphs and Ramsey Properties of Graphs and the Integers
Independent Sets in Hypergraphs and Ramsey Properties of Graphs and the Integers Open
Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris and Samotij, and independently Saxton and Thomason developed very general container…
View article: On the maximum number of integer colourings with forbidden monochromatic sums
On the maximum number of integer colourings with forbidden monochromatic sums Open
Let $f(n,r)$ denote the maximum number of colourings of $A \subseteq \lbrace 1,\ldots,n\rbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\lbrace x,y,z\rbrace$ such that $x+y=z$. We show that $f(n,2…
View article: Proof of Komlós's conjecture on Hamiltonian subsets
Proof of Komlós's conjecture on Hamiltonian subsets Open
Komlós conjectured in 1981 that among all graphs with minimum degree at least d, the complete graph Kd+1 minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this …
View article: Independent sets in hypergraphs and Ramsey properties of graphs and the\n integers
Independent sets in hypergraphs and Ramsey properties of graphs and the\n integers Open
Many important problems in combinatorics and other related areas can be\nphrased in the language of independent sets in hypergraphs. Recently Balogh,\nMorris and Samotij, and independently Saxton and Thomason developed very\ngeneral contai…
View article: On Degree Sequences Forcing The Square of a Hamilton Cycle
On Degree Sequences Forcing The Square of a Hamilton Cycle Open
A famous conjecture of Pósa from 1962 asserts that every graph on $n$ vertices and with minimum degree at least $2n/3$ contains the square of a Hamilton cycle. The conjecture was proven for large graphs in 1996 by Komlós, Sárközy and Szeme…
View article: Local conditions for exponentially many subdivisions
Local conditions for exponentially many subdivisions Open
Given a graph $F$, let $s_t(F)$ be the number of subdivisions of $F$, each with a different vertex set, which one can guarantee in a graph $G$ in which every edge lies in at least $t$ copies of $F$. In 1990, Tuza asked for which graphs $F$…