Peter Keevash
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View article: A generalised Ramsey--Turán problem for matchings
A generalised Ramsey--Turán problem for matchings Open
We prove a generalised Ramsey--Turán theorem for matchings, which (a) simultaneously generalises the Cockayne--Lorimer Theorem (Ramsey for matchings) and the Erdős--Gallai Theorem (Turán for matchings), and (b) is a generalised Turán theor…
View article: On subsets of lattice cubes avoiding affine and spherical degeneracies
On subsets of lattice cubes avoiding affine and spherical degeneracies Open
For integers $1 < k < d-1$ and $r \ge k+2$, we establish new lower bounds on the maximum number of points in $[n]^d$ such that no $r$ lie in a $k$-dimensional affine (or linear) subspace. These bounds improve on earlier results of Sudakov-…
View article: The structure of sets with cube‐avoiding sumsets
The structure of sets with cube‐avoiding sumsets Open
Suppose is a finite abelian group, is not contained in any strict coset in , and are dense subsets of such that the sumset avoids . We show that and are almost entirely contained in sets defined by a bounded number of coordinates, that is,…
View article: Dissipative particle systems on expanders
Dissipative particle systems on expanders Open
We consider a general framework for multi-type interacting particle systems on graphs, where particles move one at a time by random walk steps, different types may have different speeds, and may interact, possibly randomly, when they meet.…
View article: Turán Problems for Expanded Hypergraphs
Turán Problems for Expanded Hypergraphs Open
We obtain new results on the Turán number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity a…
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 short proof of the existence of designs
A short proof of the existence of designs Open
We give a new proof of the existence of designs, which is much shorter and gives better bounds.
View article: The structure of sets with cube-avoiding sumsets
The structure of sets with cube-avoiding sumsets Open
We prove that if $d \ge 2$ is an integer, $G$ is a finite abelian group, $Z_0$ is a subset of $G$ not contained in any strict coset in $G$, and $E_1,\dots,E_d$ are dense subsets of $G^n$ such that the sumset $E_1+\dots+E_d$ avoids $Z_0^n$ …
View article: Additive Bases: Change of Domain
Additive Bases: Change of Domain Open
We consider two questions of Ruzsa on how the minimum size of an additive basis $B$ of a given set $A$ depends on the domain of $B$. To state these questions, for an abelian group $G$ and $A \subseteq D \subseteq G$ we write $\ell_D(A) \co…
View article: The sharp doubling threshold for approximate convexity
The sharp doubling threshold for approximate convexity Open
We show for of equal volume and that if , then (up to translation) is bounded. This establishes the sharp threshold for the quantitative stability of the Brunn–Minkowski inequality recently established by Figalli, van Hintum, and Tiba, the…
View article: On the largest product-free subsets of the alternating groups
On the largest product-free subsets of the alternating groups Open
A subset $A$ of a group $G$ is called product-free if there is no solution to $a=bc$ with $a,b,c$ all in $A$ . It is easy to see that the largest product-free subset of the symmetric group $S_{n}$ is obtained by taking…
View article: Robot Positioning Using Torus Packing for Multisets
Robot Positioning Using Torus Packing for Multisets Open
We consider the design of a positioning system where a robot determines its position from local observations. This is a well-studied problem of considerable practical importance and mathematical interest. The dominant paradigm derives from…
View article: Balanced two-type annihilation: mean-field asymptotics
Balanced two-type annihilation: mean-field asymptotics Open
We consider an interacting particle system where equal-sized populations of two types of particles move by random walk steps on a graph, the two types may have different speeds, and meetings of opposite-type particles result in annihilatio…
View article: Dissipative particle systems on expanders
Dissipative particle systems on expanders Open
We consider a general framework for multi-type interacting particle systems on graphs, where particles move one at a time by random walk steps, different types may have different speeds, and may interact, possibly randomly, when they meet.…
View article: On the length of directed paths in digraphs
On the length of directed paths in digraphs Open
Thomassé conjectured the following strengthening of the well-known Caccetta-Haggkvist Conjecture: any digraph with minimum out-degree $δ$ and girth $g$ contains a directed path of length $δ(g-1)$. Bai and Manoussakis \cite{Bai} gave counte…
View article: Sharp bounds for the Tao-Vu Discrete John's Theorem
Sharp bounds for the Tao-Vu Discrete John's Theorem Open
Tao and Vu showed that every centrally symmetric convex progression $C\subset\mathbb{Z}^d$ is contained in a generalised arithmetic progression of size $d^{O(d^2)} \# C$. Berg and Henk improved the size bound to $d^{O(d\log d)} \# C$. We o…
View article: Forbidden intersections for codes
Forbidden intersections for codes Open
Determining the maximum size of a ‐intersecting code in was a longstanding open problem of Frankl and Füredi, solved independently by Ahlswede and Khachatrian and by Frankl and Tokushige. We extend their result to the setting of forbidden …
View article: Sharp hypercontractivity for symmetric groups and its applications
Sharp hypercontractivity for symmetric groups and its applications Open
A recently fertile strand of research in Group Theory is developing non-abelian analogues of classical combinatorial results for arithmetic Cayley graphs, describing properties such as growth, expansion, mixing, diameter, etc. We consider …
View article: On Ruzsa's discrete Brunn-Minkowski conjecture
On Ruzsa's discrete Brunn-Minkowski conjecture Open
We prove a conjecture by Ruzsa from 2006 on a discrete version of the Brunn-Minkowski inequality, stating that for any $A,B\subset\mathbb{Z}^k$ and $ε>0$ with $B$ not contained in $n_{k,ε}$ parallel hyperplanes we have $|A+B|^{1/k}\geq |A|…
View article: The sharp doubling threshold for approximate convexity
The sharp doubling threshold for approximate convexity Open
We show for $A,B\subset\mathbb{R}^d$ of equal volume and $t\in (0,1/2]$ that if $|tA+(1-t)B|< (1+t^d)|A|$, then (up to translation) $|\text{co}(A\cup B)|/|A|$ is bounded. This establishes the sharp threshold for Figalli and Jerison's quant…
View article: Locality in Sumsets
Locality in Sumsets Open
Motivated by the Polynomial Freiman-Ruzsa (PFR) Conjecture, we develop a theory of locality in sumsets, with applications to John-type approximation and sets with small doubling. First we show that if $A \subset \mathbb{Z}$ with $|A+A| \le…
View article: Hypercontractivity for global functions and sharp thresholds
Hypercontractivity for global functions and sharp thresholds Open
The classical hypercontractive inequality for the noise operator on the discrete cube plays a crucial role in many of the fundamental results in the Analysis of Boolean functions, such as the Kahn-Kalai-Linial theorem, Friedgut’s junta the…
View article: Isoperimetric stability in lattices
Isoperimetric stability in lattices Open
We obtain isoperimetric stability theorems for general Cayley digraphs on $\mathbb{Z}^d$. For any fixed $B$ that generates $\mathbb{Z}^d$ over $\mathbb{Z}$, we characterise the approximate structure of large sets $A$ that are approximately…
View article: Locality in sumsets
Locality in sumsets Open
Motivated by the Polynomial Freiman-Ruzsa (PFR) Conjecture, we develop a theory of locality in sumsets, with several applications to John-type approximation and stability of sets with small doubling. One highlight shows that if $A \sub \mb…
View article: The optimal edge-colouring threshold
The optimal edge-colouring threshold Open
Consider any dense r-regular quasirandom bipartite graph H with parts of size n and fix a set of r colours. Let L be a random list assignment where each colour is available for each edge of H with probability p. We show that the threshold …
View article: The existence of subspace designs
The existence of subspace designs Open
We prove the existence of subspace designs with any given parameters, provided that the dimension of the underlying space is sufficiently large in terms of the other parameters of the design and satisfies the obvious necessary divisibility…
View article: Finding matchings in dense hypergraphs
Finding matchings in dense hypergraphs Open
We consider the algorithmic decision problem that takes as input an $n$-vertex $k$-uniform hypergraph $H$ with minimum codegree at least $m-c$ and decides whether it has a matching of size $m$. We show that this decision problem is fixed p…