arXiv subjects

Katherine Staden

Publications and source records attributed to Katherine Staden.

At least 19 recordsLinked to original sources

arXiv2025

Rainbow subgraphs of star-coloured graphs

An edge-colouring of a graph $G$ can fail to be rainbow for two reasons: either it contains a monochromatic cherry (a pair of incident edges), or a monochromatic matching of size two. A colouring is a proper colouring if it forbids the first structure, and a star-colouring if it forbids the second structure. In this paper, we study rainbow subgraphs in star-coloured graphs and determine the maximum number of colours in a star-colouring of a large complete graph which does not contain a rainbow copy of a given graph $H$. This problem is a special case of one studied by Axenovich and Iverson on generalised Ramsey numbers and we extend their results in this case.

math.CO↗

arXiv2025

Geometric constructions for Ramsey-Turán theory

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 $K_q$-free graph in which every set of at least $m$ vertices contains a $K_p$. Two major open problems in this area from the 80s ask: (1) whether the asymptotic extremal structure for the general case exhibits certain periodic behaviour, resembling that of the special case when $p=2$; (2) constructing analogues of Bollobás-Erdős graphs with densities other than $1/2$. We refute the first conjecture by witnessing asymptotic extremal structures that are drastically different from the $p=2$ case, and address the second problem by constructing Bollobás-Erdős-type graphs using high dimensional complex spheres with all rational densities. Some matching upper bounds are also provided.

math.CO↗

arXiv2025

The semi-inducibility problem

Let $H$ be a $k$-edge-coloured graph and let $n$ be a positive integer. What is the maximum number of copies of $H$ in a $k$-edge-coloured complete graph on $n$ vertices? This paper studies the case $k=2$, which we call the semi-inducibility problem. This problem is a generalisation of the inducibility problem of Pippenger and Golumbic which is solved only for some small graphs and limited families of graphs. We prove sharp or almost sharp results for alternating walks, for alternating cycles of length divisible by 4, and for 4-cycles of every colour pattern. Liu, Mubayi and Reiher asked whether there is a graph $F$ for which the binomial random graph is an asymptotically extremal graph in the inducibility problem over all graphs of a given edge density. This was recently answered in a strong negative sense by Jain, Michelen and Wei. In contrast, we find a \emph{quantum} graph $Q$ with positive coefficients and an interval of edge densities for which the only extremal graphs are quasirandom.

math.CO↗

arXiv2025

A framework for the generalised Erdős-Rothschild problem and a resolution of the dichromatic triangle case

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 generalisation of this problem to a given forbidden family of colourings of $K_k$. This problem typically exhibits a dichotomy whereby for some values of $s$, the extremal graph is the `trivial' one, namely the Turán graph on $k-1$ parts, with no copies of $K_k$; while for others, this graph is no longer extremal and determining the extremal graph becomes much harder. We generalise a framework developed for the monochromatic Erdős-Rothschild problem to the general setting and work in this framework to obtain our main results, which concern two specific forbidden families: triangles with exactly two colours, and improperly coloured cliques. We essentially solve these problems fully for all integers $s \geq 2$ and large $n$. In both cases we obtain an infinite family of structures which are extremal for some $s$, which are the first results of this kind. A consequence of our results is that for every non-monochromatic colour pattern, every extremal graph is complete partite. Our work extends work of Hoppen, Lefmann and Schmidt and of Benevides, Hoppen and Sampaio.

math.CO↗

arXiv2024

Stability of transversal Hamilton cycles and paths

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 Hamilton cycles: the minimum degree required to guarantee a transversal Hamilton cycle can be lowered as long as the graph collection $G_1,\ldots,G_n$ is far in edit distance from several extremal cases. We obtain an analogous result for Hamilton paths. The proof is a combination of our newly developed regularity-blow-up method for transversals, along with the absorption method.

math.CO↗

arXiv2023

Exact solutions to the Erdős-Rothschild problem

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 edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\textbf{k})$ to denote the maximum of $F(G;\textbf{k})$ over all graphs $G$ on $n$ vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erdős and Rothschild in 1974. We prove some new exact results for $n \to \infty$: (i) A sufficient condition on $\textbf{k}$ which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results. (ii) Addressing the original question of Erdős and Rothschild, in the case $\textbf{k}=(3,\ldots,3)$ of length $7$, the unique extremal graph is the complete balanced $8$-partite graph, with colourings coming from Hadamard matrices of order $8$. (iii) In the case $\textbf{k}=(k+1,k)$, for which the sufficient condition in (i) does not hold, for $3 \leq k \leq 10$, the unique extremal graph is complete $k$-partite with one part of size less than $k$ and the other parts as equal in size as possible.

math.CO↗

arXiv2023

Universality for transversal Hamilton cycles

Let $\mathbf{G}=\{G_1, \ldots, G_m\}$ be a graph collection on a common vertex set $V$ of size $n$ such that $\delta(G_i) \geq (1+o(1))n/2$ for every $i \in [m]$. We show that $\mathbf{G}$ contains every Hamilton cycle pattern. That is, for every map $\chi: [n] \to [m]$ there is a Hamilton cycle whose $i$-th edge lies in $G_{\chi(i)}$.

math.CO↗

arXiv2023

Stability from graph symmetrisation arguments with applications to inducibility

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, it applies to the inducibility problem for an arbitrary complete bipartite graph $B$, which asks for the maximum number of induced copies of $B$ in an $n$-vertex graph, and to the inducibility problem for $K_{2,1,1,1}$ and $K_{3,1,1}$, the only complete partite graphs on at most five vertices for which the problem was previously open.

math.CO↗

arXiv2023

Transversals via regularity

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 each $G_i$. We study the case when $H$ is spanning and explore how the regularity blow-up method, that has been so successful in the uncoloured setting, can be used to find transversals. We provide the analogues of the tools required to apply this method in the transversal setting. Our main result is a blow-up lemma for transversals that applies to separable bounded degree graphs $H$. Our proofs use weak regularity in the $3$-uniform hypergraph whose edges are those $xyc$ where $xy$ is an edge in the graph $G_c$. We apply our lemma to give a large class of spanning $3$-uniform linear hypergraphs $H$ such that any sufficiently large uniformly dense $n$-vertex $3$-uniform hypergraph with minimum vertex degree $Ω(n^2)$ contains $H$ as a subhypergraph. This extends work of Lenz, Mubayi and Mycroft.

math.CO↗

arXiv2023

Stability for the Erdős-Rothschild problem

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 of colour $c$ contain no clique of order $k_c$. Write $F(n;\mathbf{k})$ to denote the maximum of $F(G;\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. In previous work with Yilma, we constructed a finite optimisation problem whose maximum is equal to the limit of $\log_2 F(n;\mathbf{k})/{n\choose 2}$ as $n$ tends to infinity and proved a stability theorem for complete multipartite graphs $G$. In this paper we provide a sufficient condition on $\mathbf{k}$ which guarantees a general stability theorem for any graph $G$, describing the asymptotic structure of $G$ on $n$ vertices with $F(G;\mathbf{k}) = F(n;\mathbf{k}) \cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem. We apply our theorem to systematically recover existing stability results as well as all cases with $s=2$. The proof uses a novel version of symmetrisation on edge-coloured weighted multigraphs.

math.CO↗

arXiv2020

The bandwidth theorem for locally dense graphs

The Bandwidth theorem of Böttcher, Schacht and Taraz gives a condition on the minimum degree of an $n$-vertex graph $G$ that ensures $G$ contains every $r$-chromatic graph $H$ on $n$ vertices of bounded degree and of bandwidth $o(n)$, thereby proving a conjecture of Bollobás and Komlós. In this paper we prove a version of the Bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense $n$-vertex graph $G$ with $δ(G) > (1/2+o(1))n$ contains as a subgraph any given (spanning) $H$ with bounded maximum degree and sublinear bandwidth.

math.CO↗

arXiv2020

The exact minimum number of triangles in graphs of given order and size

What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first non-trivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős-Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from~$1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.

math.CO↗

arXiv2020

The generalised Oberwolfach problem

We prove that any quasirandom dense large graph in which all degrees are equal and even can be decomposed into any given collection of two-factors (2-regular spanning subgraphs). A special case of this result gives a new solution to the Oberwolfach problem.

math.CO↗

arXiv2020

Ringel's tree packing conjecture in quasirandom graphs

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.

math.CO↗

arXiv2019

Minimum number of additive tuples in groups of prime order

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 $A_0,\dots,A_k\subseteq\mathbb{Z}_p$ of sizes $a_0,\dots,a_k$ respectively. An elegant argument of Lev (independently rediscovered by Samotij and Sudakov) shows that there exists an extremal configuration with all sets $A_i$ being intervals of appropriate length, and that the same conclusion also holds for the related problem, reposed by Bajnok, when $a_0=\dots=a_k=:a$ and $A_0=\dots=A_k$, provided $k$ is not equal 1 modulo $p$. By applying basic Fourier analysis, we show for Bajnok's problem that if $p\ge 13$ and $a\in\{3,\dots,p-3\}$ are fixed while $k\equiv 1\pmod p$ tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.

math.CO↗

arXiv2018

Independent sets in hypergraphs and Ramsey properties of graphs and the integers

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 theorems for independent sets in hypergraphs; both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Turán) properties of (hyper)graphs and the integers. In particular: (i) We generalise the random Ramsey theorem of Rödl and Ruciński by providing a resilience analogue. Our result unifies and generalises several fundamental results in the area including the random version of Turán's theorem due to Conlon and Gowers and Schacht. (ii) The above result also resolves a general subcase of the asymmetric random Ramsey conjecture of Kohayakawa and Kreuter. (iii) All of the above results in fact hold for uniform hypergraphs. (iv) For a (hyper)graph $H$, we determine, up to an error term in the exponent, the number of $n$-vertex (hyper)graphs $G$ that have the Ramsey property with respect to $H$ (that is, whenever $G$ is $r$-coloured, there is a monochromatic copy of $H$ in $G$). (v) We strengthen the random Rado theorem of Friedgut, Rödl and Schacht by proving a resilience version of the result. (vi) For partition regular matrices $A$ we determine, up to an error term in the exponent, the number of subsets of $\{1,\dots,n\}$ for which there exists an $r$-colouring which contains no monochromatic solutions to $Ax=0$. Along the way a number of open problems are posed.

math.CO↗

arXiv2017

On the maximum number of integer colourings with forbidden monochromatic sums

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 $f(n,2) = 2^{\lceil n/2\rceil}$, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of $f(n,r)$ for $r \leq 5$. Similar results were obtained by Hàn and Jiménez in the setting of finite abelian groups.

math.CO↗

arXiv2017

Proof of Komlós's conjecture on Hamiltonian subsets

Komlós conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large. In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify.

math.CO↗