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Peter Keevash

Publications and source records attributed to Peter Keevash.

At least 19 recordsLinked to original sources
arXiv2026

No-$(k+1)$-in-line problem for $k \geqslant 3$

What is the maximum number of points one can place in an $n \times n$ grid such that every Euclidean line contains at most $k$ points? For $k = 2$, this is the notorious no-three-in-line problem of Dudeney. In this paper, we resolve this problem for all other $k$ (and sufficiently large $n$). Namely, for $k \geqslant 3$ and sufficiently large $n$, we show that this maximum is exactly $kn$. To prove this, our key observation is that in the regime $k \geqslant 3$, the problem is dominated in a certain statistical sense by the influence of a small number of "heavy" lines with many grid points. We apply a result of Ehard-Glock-Joos on pseudorandom hypergraph matchings to construct a set of size $kn - o(n)$ with at most $k$ points on each heavy line, and then a crude deletion argument yields a no-$(k+1)$-in-line set of nearly the same size. Finally, we use a randomised switching procedure to complete the construction (building upon ideas of Simkin and Luria). Using similar ideas, we also address the no-four-on-a-circle problem of Erd\H{o}s and Purdy. Namely, we prove the existence of a set of $2n - o(n)$ points in the $n \times n$ grid such that no four of these points lie on a circle or a line, improving on the previous construction of size $n - o(n)$ due to Dong and Xu.

math.CO↗
arXiv2026

On the largest sum-free subset of the lattice cube

We determine the limiting density of the largest sum-free subset of the lattice cube $\{1,2,\ldots,n\}^d$ for all $d$, thus resolving the natural conjecture that it is constructed by two appropriate hyperplane slices. Equivalently, we show that the largest measure of a sum-free subset of the hypercube $(0,1)^d\subset \mathbb{R}^d$ is attained by $\setcond{x\in (0,1)^d}{1\leq L(x)<2}$ for some linear map $L:\mathbb{R}^d\to \mathbb{R}$. It is natural to conjecture that the same phenomenon might hold if one replaces the hypercube by any convex set not containing the origin, but we give an example to show that for sufficiently large $d$ this is not the case.

math.CO↗
arXiv2026

A very robust Ramsey theorem for matchings

Our main result is a robust generalisation of the Cockayne-Lorimer theorem on the multicolour Ramsey number of matchings. It is moreover a generalisation of the transference generalisation of Cockayne-Lorimer, which (informally) says that the random graph $G \sim G(n,p)$ with $np \to \infty$ has, with high probability, essentially the same Ramsey matching properties as the complete graph $K_n$. We show, somewhat surprisingly, that the same is true under the rather weak robustness assumption that $G$ is an $s$-connector (i.e. $\overline{G}$ is $K_{s,s}$-free) with $s=o(n)$. Moreover, we show that such $G$ has only an additive $O(s)$ loss with respect to $K_n$ for monochromatic matchings, which is essentially sharp. Our proof adapts a compression algorithm based on Gallai-Edmonds decompositions that we developed previously for generalised Ramsey-Tur\'an problems.

math.CO↗
arXiv2026

Source localisation in simple random walks

We consider the problem of locating the source (starting vertex) of a simple random walk, given a snapshot of the set of edges (or vertices) visited in the first $n$ steps. Considering lattices $\mathbb{Z}^d$, in dimensions $d \geq 5$, we show that the source can be identified (a) with probability bounded away from $0$ using one guess, and (b) with probability arbitrarily close to $1$ using a constant number of guesses. On the other hand, for dimensions $d \leq 2$, we show that one cannot locate the source with positive constant probability. Our arguments apply more generally to strongly transient and recurrent simple random walks on vertex-transitive graphs.

math.PR↗
arXiv2025

A generalised Ramsey--Tur\'an problem for matchings

We prove a generalised Ramsey--Tur\'an theorem for matchings, which (a) simultaneously generalises the Cockayne--Lorimer Theorem (Ramsey for matchings) and the Erd\H{o}s--Gallai Theorem (Tur\'an for matchings), and (b) is a generalised Tur\'an theorem in the sense that we can optimise the count of any clique (Tur\'an-type theorems optimise the count of edges). More precisely, for integers $q \ge 1$, $n \ge \ell \ge 2$, and $t_1,\dots,t_q \ge 1$ we determine the maximum number of $\ell$-vertex complete subgraphs in an $n$-vertex graph that admits a $q$-edge-colouring in which, for each $j=1,\dots,q$, the $j$-coloured subgraph has no matching of size $t_j$. We achieve this by identifying two explicit constructions and applying a compression argument to show that one of them achieves the maximum. Our compression algorithm is quite intricate and introduces methods that have not previously been applied to these types of problems: it employs an optimisation problem defined by the Gallai--Edmonds decompositions of each colour.

math.CO↗
arXiv2025

On subsets of lattice cubes avoiding affine and spherical degeneracies

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-Tomon and Lefmann. Further, we provide a randomised construction for the no-four-on-a-circle problem posed by Erd\H{o}s and Purdy, improving Thiele's bound. We also consider the random construction in higher dimensions, and improve the bound of Suk and White for $d \geq 4$. In each case, we apply the deletion method, using results from number theory and incidence geometry to solve the associated counting problems.

math.CO↗
arXiv2025

Cyclic subsets in regular Dirac graphs

In 1996, in his last paper, Erd\H{o}s asked the following question that he formulated together with Faudree: is there a positive $c$ such that any $(n+1)$-regular graph $G$ on $2n$ vertices contains at least $c 2^{2n}$ distinct vertex-subsets $S$ that are cyclic, meaning that there is a cycle in $G$ using precisely the vertices in $S$. We answer this question in the affirmative in a strong form by proving the following exact result: if $n$ is sufficiently large and $G$ minimises the number of cyclic subsets then $G$ is obtained from the complete bipartite graph $K_{n-1,n+1}$ by adding a $2$-factor (a spanning collection of vertex-disjoint cycles) within the part of size $n+1$. In particular, for $n$ large, this implies that the optimal $c$ in the problem is precisely $1/2$.

math.CO↗
arXiv2025

P\'osa rotation through a random permutation

What minimum degree of a graph $G$ on $n$ vertices guarantees that the union of $G$ and a random $2$-factor (or permutation) is with high probability Hamiltonian? Gir\~ao and Espuny D{\'\i}az showed that the answer lies in the interval $[\tfrac15 \log n, n^{3/4+o(1)}]$. We improve both the upper and lower bounds to resolve this problem asymptotically, showing that the answer is $(1+o(1))\sqrt{n\log n/2}$. Furthermore, if $G$ is assumed to be (nearly) regular then we obtain the much stronger bound that any degree growing at least polylogarithmically in $n$ is sufficient for Hamiltonicity. Our proofs use some insights from the rich theory of random permutations and a randomised version of the classical technique of P\'osa rotation adapted to multiple exposure arguments.

math.CO↗
arXiv2024

The structure of sets with cube-avoiding sumsets

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 $Z_0^n$ then $E_1, \dots, E_d$ essentially have bounded dimension. More precisely, they are almost entirely contained in sets $E_1' \times G^{I^c}, \dots, E_d' \times G^{I^c}$, where the size of $I \subset [n]$ is non-zero and independent of $n$, and $E_1',\dots,E_d'$ are subsets of $G^{I}$ such that the sumset $E_1'+\dots+E_d'$ avoids $Z_0^I$.

math.CO↗
arXiv2024

Additive Bases: Change of Domain

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 $\ell_D(A) \colon =\min \{ |B|: B \subseteq D, \ A \subseteq B+B \}$. Ruzsa asked how much larger can $\ell_{\mathbb{Z}}(A)$ be than $\ell_{\mathbb{Q}}(A)$ for $A\subset\mathbb{Z}$, and how much larger can $\ell_{\mathbb{N}}(A)$ be than $\ell_{\mathbb{Z}}(A)$ for $A\subset\mathbb{N}$. For the first question we show that if $\ell_{\mathbb{Q}}(A) = n$ then $\ell_{\mathbb{Z}}(A) \le 2n$, and that this is tight up to an additive error of at most $O(\sqrt{n})$. For the second question, we show that if $\ell_{\mathbb{Z}}(A) = n$ then $\ell_{\mathbb{N}}(A) \le O(n\log n)$, and this is tight up to the constant factor. We also consider these questions for higher order bases. Our proofs use some ideas that are unexpected in this context, including linear algebra and Diophantine approximation.

math.NT↗
arXiv2024

Robot Positioning Using Torus Packing for Multisets

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 the classical theory of de Bruijn sequences, where the robot has access to a window within a larger code and can determine its position if these windows are distinct. We propose an alternative model in which the robot has more limited observational powers, which we argue is more realistic in terms of engineering: the robot does not have access to the full pattern of colours (or letters) in the window, but only to the intensity of each colour (or the number of occurrences of each letter). This leads to a mathematically interesting problem with a different flavour to that arising in the classical paradigm, requiring new construction techniques. The parameters of our construction are optimal up to a constant factor, and computing the position requires only a constant number of arithmetic operations.

cs.DM↗
arXiv2024

Balanced two-type annihilation: mean-field asymptotics

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 annihilation. The key quantity of interest is the expected extinction time. Even for the mean-field setting of complete graphs, the correct order of magnitude was not previously known. Under essentially optimal assumptions on the starting configuration, we determine not only the order of magnitude but also the asymptotics: the expected extinction time on $K_{2n}$ is $(2+o(1))n\log n$, independently of the relative speeds of the two types.

math.PR↗
arXiv2024

Dissipative particle systems on expanders

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. We study the equilibrium time of the process, by which we mean the number of steps taken until no further interactions can occur. Under a rather general framework, we obtain high probability upper and lower bounds on the equilibrium time that match up to a constant factor and are of order $n\log n$ if there are order $n$ vertices and particles. We also obtain similar results for the balanced two-type annihilation model of chemical reactions; here, the balanced case (equal density of types) does not fit into our general framework and makes the analysis considerably more difficult. Our models do not admit any exact solution as for integrable systems or the duality approach available for some other particle systems, so we develop a variety of combinatorial tools for comparing processes in the absence of monotonicity.

math.PR↗
arXiv2024

On the length of directed paths in digraphs

Thomass\'{e} conjectured the following strengthening of the well-known Caccetta-Haggkvist Conjecture: any digraph with minimum out-degree $\delta$ and girth $g$ contains a directed path of length $\delta(g-1)$. Bai and Manoussakis \cite{Bai} gave counterexamples to Thomass\'{e}'s conjecture for every even $g\geq 4$. In this note, we first generalize their counterexamples to show that Thomass\'{e}'s conjecture is false for every $g\geq 4$. We also obtain the positive result that any digraph with minimum out-degree $\delta$ and girth $g$ contains a directed path of $2(1-\frac{2}{g})$. For small $g$ we obtain better bounds, e.g.~for $g=3$ we show that oriented graph with minimum out-degree $\delta$ contains a directed path of length $1.5\delta$. Furthermore, we show that each $d$-regular digraph with girth $g$ contains a directed path of length $\Omega(dg/\log d)$. Our results give the first non-trivial bounds for these problems.

math.CO↗
arXiv2024

Long induced paths in expanders

We prove that any bounded degree regular graph with sufficiently strong spectral expansion contains an induced path of linear length. This is the first such result for expanders, strengthening an analogous result in the random setting by Dragani\'c, Glock and Krivelevich. More generally, we find long induced paths in sparse graphs that satisfy a mild upper-uniformity edge-distribution condition.

math.CO↗
arXiv2023

Sharp bounds for the Tao-Vu Discrete John's Theorem

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 obtain the bound $d^{O(d)} \# C$, which is sharp up to the implied constant, and is of the same form as the bound in the continuous setting given by John's Theorem.

math.CO↗
arXiv2023

Sharp hypercontractivity for symmetric groups and its applications

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 these problems for the symmetric and alternating groups. The case of normal Cayley graphs (those generated by unions of conjugacy classes) has seen significant progress via character theory (whereby Larsen and Shalev resolved several open problems), but the general case still remains poorly understood. In this paper we generalise the background assumption from being normal to being global (a pseudorandomness condition), replacing character bounds by spectral estimates for convolution operators of global functions, thus obtaining qualitative generalisations of several results on normal Cayley graphs. Furthermore, our theory in the pseudorandom setting can be applied (via density increment arguments) to several results for general sets that are not too sparse, including analogues of Polynomial Freiman-Ruzsa, Bogolyubov's lemma, Roth's theorem, the Waring problem and essentially sharp estimates for the diameter problem of Cayley graphs whose density is at least exponential in -n. Our main tool is a sharp new hypercontractive inequality for global functions on the symmetric group.

math.GR↗