On Kahn's flow conjecture
We prove Kahn's flow conjecture, which is a strong form of Chvátal's conjecture on the largest intersecting subfamily of a downset.
math.CO↗
arXiv subjects
Peter Keevash
Publications and source records attributed to Peter Keevash.
A non-trivial bound for 3AP-intersecting families
A family $F$ of subsets of $[n]$ is 3AP-intersecting if every two members have intersection containing a non-trivial three-term arithmetic progression. We prove that there is an absolute constant $c>0$ such that any such $F$ has size at most $(\tfrac12 - c)2^n$. This is the first non-trivial progress towards a conjecture of Simonovits and Sós that the maximum possible size is $2^{n-3}$. More generally, we show the same bound for $H$-intersecting families whenever $H$ is a $3$-graph on $[n]$ with bounded codegrees. A clique shows that this is sharp, in that the bounded codegree assumption cannot be removed.
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Finding matchings in dense hypergraphs
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 parameter tractable with respect to $c$. Furthermore, our algorithm not only decides the problem, but actually either finds a matching of size $m$ or a certificate that no such matching exists. In particular, when $m=n/k$ and $c=O(\log n)$, this gives a polynomial-time algorithm, that given any $n$-vertex $k$-uniform hypergraph $H$ with minimum codegree at least $n/k-c$, finds either a perfect matching in $H$ or a certificate that no perfect matching exists.
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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 Erdős 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.
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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.
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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.
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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-Turán problems.
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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.
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A generalised Ramsey--Turán problem for matchings
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 theorem in the sense that we can optimise the count of any clique (Turán-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.
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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 Erdős 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.
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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.
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Cyclic subsets in regular Dirac graphs
In 1996, in his last paper, Erdős 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 $1/2$.
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Pósa 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? Girão and Espuny D{\'ı}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 Pósa rotation adapted to multiple exposure arguments.
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The existence of designs
We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition. As a further generalisation, we obtain the same conclusion only assuming an extendability property and the existence of a robust fractional clique decomposition.
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A short proof of the existence of designs
We give a new proof of the existence of designs, which is much shorter and gives better bounds.
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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 $Z_0^I$.
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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 Draganić, Glock and Krivelevich. More generally, we find long induced paths in sparse graphs that satisfy a mild upper-uniformity edge-distribution condition.
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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.
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