arXiv subjects
Publications and source records attributed to Christian Reiher.
We extend the hypergraph regularity method to sparse $3$-uniform hypergraphs whose vertex links are $C_4$-free. In other words, we consider hypergraphs $H=(V,E)$ that are $K_{1,2,2}$-free, which implies that $|E|=O(|V|^{5/2})$. For such hypergraphs we establish a sparse analogue of the removal lemma for the tight cycle on seven vertices minus an edge.
Extending earlier results of Ne\v{s}et\v{r}il and R\"odl [Selective graphs and hypergraphs, Ann. Discrete Math. 3 (1978), 181--189], we show that for every ordered graph $F$ there exist an ordered graph $H$ and a system $\mathscr{H}_F$ of induced copies of $F$ such that every colouring of the edges of $H$ yields a canonically coloured copy of $F$ from $\mathscr{H}_F$ and any two copies from $\mathscr{H}_F$ intersect either in a vertex or an edge or not at all. As a consequence, this allows us to construct, for any given ordered graph $F$, canonical Ramsey graphs $H$ enjoying additional structural properties. In particular, $H$ can have the same clique number as $F$ and, provided $F$ is not bipartite, the same odd girth. Moreover, if $F$ is connected, then the copies of $F$ from $\mathscr{H}_F$ are not only induced, but their pairs of vertices also have the same distances in $H$ as in $F$.
Coherence in a monoidal category asserts that all morphisms built from structural isomorphisms with a fixed source and target coincide. These structural isomorphisms include, in particular, the associators. Linearly distributive categories carry two tensor products, with structural morphisms given by associators and distributors relating the two tensor products. In several examples, including Grothendieck--Verdier categories, also known as $\ast$-autonomous categories, these distributors need not be invertible. We give a self-contained proof that linearly distributive categories without units are coherent, while units may obstruct coherence. With the same techniques, we also establish an analogous coherence result for Frobenius linearly distributive functors without units. These results admit a reformulation in terms of directed paths in associahedra and multiplihedra.
Answering a question of Leo Versteegen, we prove that for $n\ge 3$ every sum-free set $A\subseteq\mathbb{F}_5^n$ with $|A|\ge 28\cdot 5^{n-3}$ is either contained in the union of two parallel hyperplanes, or isomorphic to $\Lambda\times \mathbb{F}_5^{n-3}$, where $\Lambda\subseteq \mathbb{F}_5^3$ denotes a certain sum-free set of size $28$ discovered by Vsevolod Lev and Leo Versteegen.
A randomly perturbed graph $G^p = G_\alpha \cup G_{n,p}$ is obtained by taking a deterministic $n$-vertex graph $G_\alpha = (V, E)$ with minimum degree $\delta(G)\geq \alpha n$ and adding the edges of the binomial random graph $G_{n,p}$ defined on the same vertex set $V$. For which value $p$ (depending on $\alpha$) does the graph $G^p$ contain a $K_r$-factor -- a spanning collection of vertex-disjoint copies of $K_r$ -- with high probability? The order of magnitude of the minimum such $p$ was determined whenever $\alpha \neq 1- \frac{s}{r}$ for an integer $s$ by Balogh, Treglown and Wagner, and by Han, Morris and Treglown. In earlier work, the first three authors determined this threshold probability $p_s$ up to a constant factor for all values of $\alpha = 1-\frac{s}{r}\leq \frac 12$. Here, we complete the picture by establishing $p_s$ in the remaining case $\alpha > \frac12$. A key ingredient in our approach is an extremal result of independent interest: we prove a fractional stability version of a tiling theorem due to Shokoufandeh and Zhao.
We study subgraphs that appear in large Ramsey graphs for a given graph $F$. The recent girth Ramsey theorem of the first two authors asserts that there are Ramsey graphs such that all small subgraphs are `forests of copies of $F$' amalgamated on vertices and edges. We derive a few further consequences from this structural result and investigate to which extent such forests of copies must be present in Ramsey graphs.
We introduce a modification of the Tur\'an density of ordered graphs and investigate this graph parameter.
Given $d\in\mathbb{N}$, let $\alpha(d)$ be the largest real number such that every abstract simplicial complex $\mathcal{S}$ with $0<\vert\mathcal{S}\vert\leq\alpha(d)\vert V(\mathcal{S})\vert$ has a vertex of degree at most $d$. We extend previous results by Frankl, Frankl and Watanabe, and Piga and Sch\"ulke by proving that for all integers $d$ and $m$ with $d\geq m\geq 1$, we have $\alpha(2^d-m)=\frac{2^{d+1}-m}{d+1}$. Similar results were obtained independently by Li, Ma, and Rong.
For a bounded measurable set $A\subseteq \mathbb{R}$ we denote the Lebesgue measure of $\{(x, y)\in A^2\colon x\le y\le x+1\}$ by $\Phi(A)$. We prove that if $I=A_1\cup\dots\cup A_{k+1}$ partitions an interval $I$ of length $L$ into $k+1$ measurable pieces, then $\sum_{i=1}^{k+1} \Phi(A_i)\ge (\sqrt{k^2+1}-k)L-1$, where the multiplicative constant $\sqrt{k^2+1}-k$ is optimal. As a matter of fact we obtain the more general result that $\Phi(A)\ge (\xi+\sqrt{1-2\xi+2\xi^2}-1)L-1$ whenever $A\subseteq I$ has measure $\xi L$.
A randomly perturbed graph $G^p = G_\alpha \cup G(n,p)$ is obtained by taking a deterministic $n$-vertex graph $G_\alpha = (V, E)$ with minimum degree $\delta(G)\geq \alpha n$ and adding the edges of the binomial random graph $G(n,p)$ defined on the same vertex set $V$. For which value $p$ (depending on $\alpha$) does the graph $G^p$ contain a $K_r$-factor (a spanning collection of vertex-disjoint $K_r$-copies) with high probability? The order of magnitude of the minimal value of $p$ has been determined whenever $\alpha \neq 1- \frac{s}{r}$ for an integer $s$ (see Han, Morris, and Treglown [RSA, 2021] and Balogh, Treglown, and Wagner [CPC, 2019]). We establish the minimal probability $p_s$ (up to a constant factor) for all values of $\alpha = 1-\frac{s}{r} \leq \frac 12$, and show that the threshold exhibits a polynomial jump at $\alpha = 1-\frac{s}{r}$ compared to the surrounding intervals. An extremal example $G_{\alpha}$ which shows that $p_s$ is optimal up to a constant factor differs from the previous (usually multipartite) examples in containing a pseudorandom induced subgraph.
We establish the existence of generalised Sidon sets enjoying additional Ramsey-type properties, which are motivated by questions of Erd\H{o}s and Newman and of Alon and Erd\H{o}s.
We classify the sum-free subsets of ${\mathbb F}_3^n$ whose density exceeds $\frac16$. This yields a resolution of Vsevolod Lev's periodicity conjecture, which asserts that if a sum-free subset ${A\subseteq {\mathbb F}_3^n}$ is maximal with respect to inclusion and aperiodic (in the sense that there is no non-zero vector $v$ satisfying $A+v=A$), then $|A|\le \frac12(3^{n-1}+1)$ -- a bound known to be optimal if $n\ne 2$, while for $n=2$ there are no such sets.
Let $p$ be a prime number with $p\equiv 2\pmod{3}$ and let $n\ge 1$ be a dimension. It is known that a sum-free subset of ${\mathbb F}_p^n$ can have at most the size $\frac13(p+1)p^{n-1}$ and that, up to automorphisms of ${\mathbb F}_p^n$, the only extremal example is the `cuboid' $\bigl[\frac{p+1}3, \frac{2p-1}3\bigr]\times {\mathbb F}_p^{n-1}$. For $p\ge 11$ we show that if a sum-free subset of ${\mathbb F}_p^n$ is not contained in such an extremal one, then its size is at most $\frac13(p-2)p^{n-1}$. This bound is optimal and we classify the extremal configurations. The remaining cases $p=2, 5$ are known to behave differently. For $p=3$ the analogous question was solved by Vsevolod Lev, and for $p\equiv 1\pmod{3}$ it is less interesting.
Solving a long standing conjecture of Erd\H{o}s and Simonovits, Brandt and Thomass\'e proved that the chromatic number of each triangle-free graph $G$ such that $\delta(G)>|V(G)|/3$ is at most four. In fact, they showed the much stronger result that every maximal triangle-free graph $G$ satisfying this minimum degree condition is a blow-up of either an Andr\'asfai or a Vega graph. Here we establish the same structural conclusion on $G$ under the weaker assumption that for $m\in\{2, 3, 4\}$ every sequence of $3m$ vertices has a subsequence of length $m+1$ with a common neighbour. In forthcoming work this will be used to solve an old problem of Andr\'asfai in Ramsey-Tur\'an theory.
This survey on graphs of large girth consists of two parts. The first deals with some aspects of algebraic and extremal graph theory loosely related to the Moore bound. Our point of departure for the second, Ramsey theoretic, part are some constructions of graphs with large chromatic number and large girth; this will lead us to a discussion of the recent girth Ramsey theorem. Both parts can be enjoyed independently of each other.
We show that $3$-uniform hypergraphs with the property that all vertices have a quasirandom link graph with density bigger than $1/3$ contain a clique on five vertices. This result is asymptotically best possible.
Erdős and Hajnal proved that every graph of uncountable chromatic number contains arbitrarily large finite, complete, bipartite graphs. We extend this result to hypergraphs.
We construct for every integer $k\geq 3$ and every real $\mu\in(0, \frac{k-1}{k})$ a set of integers $X=X(k, \mu)$ which, when coloured with finitely many colours, contains a monochromatic $k$-term arithmetic progression, whilst every finite $Y\subseteq X$ has a subset $Z\subseteq Y$ of size $|Z|\geq \mu |Y|$ that is free of arithmetic progressions of length $k$. This answers a question of Erd\H{o}s, Ne\v{s}et\v{r}il, and the second author. Moreover, we obtain an analogous multidimensional statement and a Hales-Jewett version of this result.