arXiv subjects

Eoin Long

Publications and source records attributed to Eoin Long.

At least 19 recordsLinked to original sources

arXiv2024

Distinct degrees and homogeneous sets II

Given an $n$-vertex graph $G$, let $\hom (G)$ denote the size of a largest homogeneous set in $G$ and let $f(G)$ denote the maximal number of distinct degrees appearing in an induced subgraph of $G$. The relationship between these parameters has been well studied by several researchers over the last 40 years, beginning with Erd\H{o}s, Faudree and S\'os in the Ramsey regime when $\hom (G) = O(\log n)$. Our main result here proves that any $n$-vertex graph $G$ with $\hom (G) \leq n^{1/2}$ satisfies \begin{align*} f(G) \geq \sqrt[3]{\frac {n^2}{\hom (G)} } \cdot n^{-o(1)}. \end{align*} This confirms a conjecture of the authors from a previous work, in which we addressed the $\hom (G) \geq n^{1/2}$ regime. Together, these provide the complete extremal relationship between these parameters (asymptotically), showing that any $n$-vertex graph $G$ satisfies \begin{align*} \max \Big ( f(G) \cdot \hom (G), \sqrt {f(G) ^3 \cdot \hom (G) } \Big ) \geq n^{1-o(1)}. \end{align*} This relationship is tight (up to the $n^{-o(1)}$ term) for all possible values of $\hom (G)$, from $\Omega (\log n )$ to $n$, as demonstrated by appropriately generated Erd\H{o}s $-$ Renyi random graphs.

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arXiv2022

A bipartite version of the Erd\H{o}s $-$ McKay conjecture

An old conjecture of Erd\H{o}s and McKay states that if all homogeneous sets in an $n$-vertex graph are of order $O(\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega (n^2)\}$. We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n \times n$ bipartite graph are of order $O(\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega (n^2)\}$.

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arXiv2022

Distinct degrees and homogeneous sets

In this paper we investigate the extremal relationship between two well-studied graph parameters: the order of the largest homogeneous set in a graph $G$ and the maximal number of distinct degrees appearing in an induced subgraph of $G$, denoted respectively by $\hom (G)$ and $f(G)$. Our main theorem improves estimates due to several earlier researchers and shows that if $G$ is an $n$-vertex graph with $\hom (G) \geq n^{1/2}$ then $f(G) \geq \big ( {n}/{\hom (G)} \big )^{1 - o(1)}$. The bound here is sharp up to the $o(1)$-term, and asymptotically solves a conjecture of Narayanan and Tomon. In particular, this implies that $\max \{ \hom (G), f(G) \} \geq n^{1/2 -o(1)}$ for any $n$-vertex graph $G$,which is also sharp. The above relationship between $\hom (G)$ and $f(G)$ breaks down in the regime where $\hom (G) < n^{1/2}$. Our second result provides a sharp bound for distinct degrees in biased random graphs, i.e. on $f\big (G(n,p) \big )$. We believe that the behaviour here determines the extremal relationship between $\hom (G)$ and $f(G)$ in this second regime. Our approach to lower bounding $f(G)$ proceeds via a translation into an (almost) equivalent probabilistic problem, and it can be shown to be effective for arbitrary graphs. It may be of independent interest.

math.CO↗

arXiv2021

Global hypercontractivity and its applications

The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedgut's junta theorem and the invariance principle. In these results the cube is equipped with the uniform measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general $p$-biased measures. However, simple examples show that when $p = o(1)$, there is no hypercontractive inequality that is strong enough. In this paper, we establish an effective hypercontractive inequality for general $p$ that applies to `global functions', i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain's sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain's theorem, thereby making progress on a conjecture of Kahn and Kalai and by establishing a $p$-biased analog of the invariance principle. Our results have significant applications in Extremal Combinatorics. Here we obtain new results on the Tur\'an 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 and the number of edges and solve a number of open problems in the area. In particular, we give general conditions under which the crosscut parameter asymptotically determines the Tur\'an number, answering a question of Mubayi and Verstra\"ete. We also apply the Junta Method to refine our asymptotic results and obtain several exact results, including proofs of the Huang--Loh--Sudakov conjecture on cross matchings and the F\"uredi--Jiang--Seiver conjecture on path expansions.

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arXiv2021

Forbidden intersections for codes

Determining the maximum size of a $t$-intersecting code in $[m]^n$ was a longstanding open problem of Frankl and F\"uredi, solved independently by Ahlswede and Khachatrian and by Frankl and Tokushige. We extend their result to the setting of forbidden intersections, by showing that for any $m>2$ and $n$ large compared with $t$ (but not necessarily $m$) that the same bound holds for codes with the weaker property of being $(t-1)$-avoiding, i.e.\ having no two vectors that agree on exactly $t-1$ coordinates. Our proof proceeds via a junta approximation result of independent interest, which we prove via a development of our recent theory of global hypercontractivity: we show that any $(t-1)$-avoiding code is approximately contained in a $t$-intersecting junta (a code where membership is determined by a constant number of co-ordinates). In particular, when $t=1$ this gives an alternative proof of a recent result of Eberhard, Kahn, Narayanan and Spirkl that symmetric intersecting codes in $[m]^n$ have size $o(m^n)$.

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arXiv2019

Distinct degrees in induced subgraphs

An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An $N$-vertex graph is called $C$-Ramsey if it has no homogeneous set of size $C\log N$. A theorem of Bukh and Sudakov, solving a conjecture of Erd\H{o}s, Faudree and S\'os, shows that any $C$-Ramsey $N$-vertex graph contains an induced subgraph with $\Omega_C(N^{1/2})$ distinct degrees. We improve this to $\Omega_C(N^{2/3})$, which is tight up to the constant factor. We also show that any $N$-vertex graph with $N > (k-1)(n-1)$ and $n\geq n_0(k) = \Omega (k^9)$ either contains a homogeneous set of order $n$ or an induced subgraph with $k$ distinct degrees. The lower bound on $N$ here is sharp, as shown by an appropriate Tur\'an graph, and confirms a conjecture of Narayanan and Tomon.

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arXiv2019

Correlation for permutations

In this note we investigate correlation inequalities for `up-sets' of permutations, in the spirit of the Harris--Kleitman inequality. We focus on two well-studied partial orders on $S_n$, giving rise to differing notions of up-sets. Our first result shows that, under the strong Bruhat order on $S_n$, up-sets are positively correlated (in the Harris--Kleitman sense). Thus, for example, for a (uniformly) random permutation $\pi$, the event that no point is displaced by more than a fixed distance $d$ and the event that $\pi$ is the product of at most $k$ adjacent transpositions are positively correlated. In contrast, under the weak Bruhat order we show that this completely fails: surprisingly, there are two up-sets each of measure $1/2$ whose intersection has arbitrarily small measure. We also prove analogous correlation results for a class of non-uniform measures, which includes the Mallows measures. Some applications and open problems are discussed.

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arXiv2019

Hypercontractivity for global functions and sharp thresholds

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 KKL (Kahn-Kalai-Linial) theorem, Friedgut's junta theorem and the invariance principle of Mossel, O'Donnell and Oleszkiewicz. In these results the cube is equipped with the uniform ($1/2$-biased) measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general $p$-biased measures. However, simple examples show that when $p$ is small there is no hypercontractive inequality that is strong enough for such applications. In this paper, we establish an effective hypercontractivity inequality for general $p$ that applies to `global functions', i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain's sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain's theorem, thereby making progress on a conjecture of Kahn and Kalai. An additional application of our hypercontractivity theorem, is a $p$-biased analog of the seminal invariance principle of Mossel, O'Donnell, and Oleszkiewicz. In a companion paper, we give applications to the solution of two open problems in Extremal Combinatorics.

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arXiv2018

Stability for vertex isoperimetry in the cube

We prove a stability version of Harper's cube vertex isoperimetric inequality, showing that subsets of the cube with vertex boundary close to the minimum possible are close to (generalised) Hamming balls. Furthermore, we obtain a local stability result for ball-like sets that gives a sharp estimate for the vertex boundary in terms of the distance from a ball, and so our stability result is essentially tight (modulo a non-monotonicity phenomenon). We also give similar results for the Kruskal--Katona Theorem and applications to new stability versions of some other results in Extremal Combinatorics.

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arXiv2018

Cycle-complete Ramsey numbers

The Ramsey number $r(C_{\ell},K_n)$ is the smallest natural number $N$ such that every red/blue edge-colouring of a clique of order $N$ contains a red cycle of length $\ell$ or a blue clique of order $n$. In 1978, Erd\H{o}s, Faudree, Rousseau and Schelp conjectured that $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ for $\ell \geq n\geq 3$ provided $(\ell,n) \neq (3,3)$. We prove that, for some absolute constant $C\ge 1$, we have $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ provided $\ell \geq C\frac {\log n}{\log \log n}$. Up to the value of $C$ this is tight since we also show that, for any $\varepsilon >0$ and $n> n_0(\varepsilon )$, we have $r(C_{\ell }, K_n) \gg (\ell -1)(n-1)+1$ for all $3 \leq \ell \leq (1-\varepsilon )\frac {\log n}{\log \log n}$. This proves the conjecture of Erd\H{o}s, Faudree, Rousseau and Schelp for large $\ell $, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erd\H{o}s, Faudree, Rousseau and Schelp.

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arXiv2017

Forbidden vector-valued intersections

We solve a generalised form of a conjecture of Kalai motivated by attempts to improve the bounds for Borsuk's problem. The conjecture can be roughly understood as asking for an analogue of the Frankl-Rödl forbidden intersection theorem in which set intersections are vector-valued. We discover that the vector world is richer in surprising ways: in particular, Kalai's conjecture is false, but we prove a corrected statement that is essentially best possible, and applies to a considerably more general setting. Our methods include the use of maximum entropy measures, VC-dimension, Dependent Random Choice and a new correlation inequality for product measures.

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arXiv2017

A stability result for the cube edge isoperimetric inequality

We prove the following stability version of the edge isoperimetric inequality for the cube: any subset of the cube with average boundary degree within $K$ of the minimum possible is $\varepsilon $-close to a union of $L$ disjoint cubes, where $L \leq L(K,\varepsilon )$ is independent of the dimension. This extends a stability result of Ellis, and can viewed as a dimension-free version of Friedgut's junta theorem.

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arXiv2016

Counting Hamilton decompositions of oriented graphs

A Hamilton cycle in a directed graph $G$ is a cycle that passes through every vertex of $G$. A Hamiltonian decomposition of $G$ is a partition of its edge set into disjoint Hamilton cycles. In the late $60$s Kelly conjectured that every regular tournament has a Hamilton decomposition. This conjecture was recently settled by Kühn and Osthus, who proved more generally that every $r$-regular $n$-vertex oriented graph $G$ (without antiparallel edges) with $r=cn$ for some fixed $c>3/8$ has a Hamiltonian decomposition, provided $n=n(c)$ is sufficiently large. In this paper we address the natural question of estimating the number of such decompositions of $G$ and show that this number is $n^{(1-o(1))cn^2}$. In addition, we also obtain a new and much simpler proof for the approximate version of Kelly's conjecture.

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arXiv2016

Packing and counting arbitrary Hamilton cycles in random digraphs

We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in ${\cal D}(n,p)$ for nearly optimal $p$ (up to a $\log ^cn$ factor). In particular, we show that given $t = (1-o(1))np$ Hamilton cycles $C_1,\ldots ,C_{t}$, each of which is oriented arbitrarily, a digraph $D \sim {\cal D}(n,p)$ w.h.p. contains edge disjoint copies of $C_1,\ldots ,C_t$, provided $p=ω(\log ^3 n/n)$. We also show that given an arbitrarily oriented $n$-vertex cycle $C$, a random digraph $D \sim {\cal D}(n,p)$ w.h.p. contains $(1\pm o(1))n!p^n$ copies of $C$, provided $p \geq \log ^{1 + o(1)}n/n$.

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arXiv2016

On the normalized Shannon capacity of a union

Let $G_1 \times G_2$ denote the strong product of graphs $G_1$ and $G_2$, i.e. the graph on $V(G_1) \times V(G_2)$ in which $(u_1,u_2)$ and $(v_1,v_2)$ are adjacent if for each $i=1,2$ we have $u_i=v_i$ or $u_iv_i \in E(G_i)$. The Shannon capacity of $G$ is $c(G) = \lim_{n\to \infty} α(G^n)^{1/n}$, where $G^n$ denotes the $n$-fold strong power of $G$, and $α(H)$ denotes the independence number of a graph $H$. The normalized Shannon capacity of $G$ is $C(G) = \frac {\log c(G)}{\log |V(G)|}$. Alon asked whether for every $ε> 0$ there are graphs $G$ and $G'$ satisfying $C(G), C(G') < ε$ but with $C(G + G') > 1 - ε$. We show that the answer is no.

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arXiv2016

Large unavoidable subtournaments

Let $D_k$ denote the tournament on $3k$ vertices consisting of three disjoint vertex classes $V_1, V_2$ and $V_3$ of size $k$, each of which is oriented as a transitive subtournament, and with edges directed from $V_1$ to $V_2$, from $V_2$ to $V_3$ and from $V_3$ to $V_1$. Fox and Sudakov proved that given a natural number $k$ and $ε> 0$ there is $n_0(k,ε)$ such that every tournament of order $n_0(k,ε)$ which is $ε$-far from being transitive contains $D_k$ as a subtournament. Their proof showed that $n_0(k,ε) \leq ε^{-O(k/ε^2)}$ and they conjectured that this could be reduced to $n_0(k,ε) \leq ε^{-O(k)}$. Here we prove this conjecture.

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arXiv2016

On a Ramsey-type problem of Erdős and Pach

In this paper we show that there exists a constant $C>0$ such that for any graph $G$ on $Ck\ln k$ vertices either $G$ or its complement $\bar{G}$ has an induced subgraph on $k$ vertices with minimum degree at least $\frac12(k-1)$. This affirmatively answers a question of Erdős and Pach from 1983.

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arXiv2015

Packing, Counting and Covering Hamilton cycles in random directed graphs

A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this, is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so called Pos?a `rotation-extension' technique for the undirected analogue. Let ${\mathcal D}(n,p)$ denote the random digraph on vertex set $[n]$, obtained by adding each directed edge independently with probability $p$. Here, we present a general and a very simple method, using known results, to attack problems of packing and counting Hamilton cycles in random directed graphs, for every edge-probability $p>\log^C(n)/n$. Our results are asymptotically optimal with respect to all parameters and apply equally well to the undirected case.

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