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Peter Keevash

Publications and source records attributed to Peter Keevash.

80 records · Page 5Linked to original sources
arXiv2009

The early evolution of the H-free process

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as $n \to \infty$, the minimum degree in G is at least $cn^{1-(v_H-2)/(e_H-1)}(\log n)^{1/(e_H-1)}$. This gives new lower bounds for the Turán numbers of certain bipartite graphs, such as the complete bipartite graphs $K_{r,r}$ with $r \ge 5$. When H is a complete graph $K_s$ with $s \ge 5$ we show that for some C>0, with high probability the independence number of G is at most $Cn^{2/(s+1)}(\log n)^{1-1/(e_H-1)}$. This gives new lower bounds for Ramsey numbers R(s,t) for fixed $s \ge 5$ and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.

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arXiv2009

Pancyclicity of Hamiltonian and highly connected graphs

A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\ell$ for all $3 \le \ell \le n$. Write $α(G)$ for the independence number of $G$, i.e. the size of the largest subset of the vertex set that does not contain an edge, and $κ(G)$ for the (vertex) connectivity, i.e. the size of the smallest subset of the vertex set that can be deleted to obtain a disconnected graph. A celebrated theorem of Chvátal and Erdős says that $G$ is Hamiltonian if $κ(G) \ge α(G)$. Moreover, Bondy suggested that almost any non-trivial conditions for Hamiltonicity of a graph should also imply pancyclicity. Motivated by this, we prove that if $κ(G) \ge 600α(G)$ then G is pancyclic. This establishes a conjecture of Jackson and Ordaz up to a constant factor. Moreover, we obtain the more general result that if G is Hamiltonian with minimum degree $δ(G) \ge 600α(G)$ then G is pancyclic. Improving an old result of Erdős, we also show that G is pancyclic if it is Hamiltonian and $n \ge 150α(G)^3$. Our arguments use the following theorem of independent interest on cycle lengths in graphs: if $δ(G) \ge 300α(G)$ then G contains a cycle of length $\ell$ for all $3 \le \ell \le δ(G)/81$.

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arXiv2008

Directed graphs without short cycles

For a directed graph $G$ without loops or parallel edges, let $β(G)$ denote the size of the smallest feedback arc set, i.e., the smallest subset $X \subset E(G)$ such that $G \sm X$ has no directed cycles. Let $γ(G)$ be the number of unordered pairs of vertices of $G$ which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least $r \ge 4$ satisfies $β(G) \le cγ(G)/r^2$, where $c$ is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour, and Sullivan. This result can be also used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed $0 < θ< 1/2$ and sufficiently large $n$, if $G$ is a digraph with $n$ vertices and $β(G) \ge θn^2$, then for any $0 \le m \le θn-o(n)$ it contains a directed cycle whose length is between $m$ and $m+6 θ^{-1/2}$. Moreover, there is a constant $C$ such that either $G$ contains directed cycles of every length between $C$ and $θn-o(n)$ or it is close to a digraph $G'$ with a simple structure: every strong component of $G'$ is periodic. These results are also tight up to the constant factors.

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arXiv2008

A hypergraph regularity method for generalised Turan problems

We describe a method that we believe may be foundational for a comprehensive theory of generalised Turan problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that any 3-graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a Fano plane, for n sufficiently large. For projective planes over fields of odd size q we show that the codegree threshold is between n/2-q+1 and n/2, but for PG_2(4) we find the somewhat surprising phenomenon that the threshold is less than (1/2-c)n for some small c>0. We conclude by setting out a program for future developments of this method to tackle other problems.

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arXiv2008

Triangle packings and 1-factors in oriented graphs

An oriented graph is a directed graph which can be obtained from a simple undirected graph by orienting its edges. In this paper we show that any oriented graph G on n vertices with minimum indegree and outdegree at least (1/2-o(1))n contains a packing of cyclic triangles covering all but at most 3 vertices. This almost answers a question of Cuckler and Yuster and is best possible, since for n = 3 mod 18 there is a tournament with no perfect triangle packing and with all indegrees and outdegrees (n-1)/2 or (n-1)/2 \pm 1. Under the same hypotheses, we also show that one can embed any prescribed almost 1-factor, i.e. for any sequence n_1,...,n_t with n_1+...+n_t < n-O(1) we can find a vertex-disjoint collection of directed cycles with lengths n_1,...,n_t. In addition, under quite general conditions on the n_i we can remove the O(1) additive error and find a prescribed 1-factor.

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arXiv2008

Shadows and intersections: stability and new proofs

We give a short new proof of a version of the Kruskal-Katona theorem due to Lovász. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation of Lovász's theorem that answers a question of Frankl and Tokushige.

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arXiv2002

On a hypergraph Turan problem of Frankl

Let $C^{2k}_r$ be the $2k$-uniform hypergraph obtained by letting $P_1,...,P_r$ be pairwise disjoint sets of size $k$ and taking as edges all sets $P_i \cup P_j$ with $i \neq j$. This can be thought of as the `$k$-expansion' of the complete graph $K_r$: each vertex has been replaced with a set of size $k$. We determine the exact Turan number of $C^{2k}_3$ and the corresponding extremal hypergraph, thus confirming a conjecture of Frankl. Sidorenko has given an upper bound of $(r-2) / (r-1)$ for the Turán density of $C^{2k}_r$ for any $r$, and a construction establishing a matching lower bound when $r$ is of the form $2^p + 1$. We show that when $r = 2^p + 1$, any $C^4_r$-free hypergraph of density $(r-2)/(r-1) - o(1)$ looks approximately like Sidorenko's construction. On the other hand, when $r$ is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density of $C^4_r$ to $(r-2)/(r-1) - c(r)$, where $c(r)$ is a constant depending only on $r$. The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal-Katona theorem and properties of Krawtchouck polynomials.

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