Proceedings of the London Mathematical Society, ISSN 1460-244X, 2018, Volume 118, Issue 6, pp. 1393 - 1449

We provide a degree condition on a regular n‐vertex graph G which ensures the existence of a near optimal packing of any family H of bounded degree n‐vertex...

05D40 (primary) | 05C70

05D40 (primary) | 05C70

Journal Article

The Bulletin of the London Mathematical Society, ISSN 0024-6093, 2017, Volume 49, Issue 6, pp. 991 - 999

We affirmatively answer a question of Erdős and Pach from 1983 by showing the following: there is some constant C>0 such that for any graph G on Cklnk vertices...

05C55 (primary) | 05D40 (secondary) | 05D10

05C55 (primary) | 05D40 (secondary) | 05D10

Journal Article

Combinatorica, ISSN 0209-9683, 8/2019, Volume 39, Issue 4, pp. 911 - 921

Journal Article

Bulletin of the London Mathematical Society, ISSN 0024-6093, 08/2017, Volume 49, Issue 4, pp. 745 - 747

Folkman's theorem asserts that for each k∈N, there exists a natural number n=F(k) such that whenever the elements of [n] are two‐coloured, there exists a set...

05D40 (secondary) | 05D10 (primary) | MATHEMATICS | Mathematics - Combinatorics

05D40 (secondary) | 05D10 (primary) | MATHEMATICS | Mathematics - Combinatorics

Journal Article

Journal of the London Mathematical Society, ISSN 0024-6107, 06/2019, Volume 99, Issue 3, pp. 757 - 777

Consider integers k,ℓ such that 0⩽ℓ⩽k2. Given a large graph G, what is the fraction of k‐vertex subsets of G which span exactly ℓ edges? When G is empty or...

60C05 (secondary) | 05C35 | 05D40 (primary) | MATHEMATICS | MATRICES | SIZES | SMALL NUMBER | SINGULARITY | INDUCIBILITY | GRAPHS | Mathematics - Combinatorics

60C05 (secondary) | 05C35 | 05D40 (primary) | MATHEMATICS | MATRICES | SIZES | SMALL NUMBER | SINGULARITY | INDUCIBILITY | GRAPHS | Mathematics - Combinatorics

Journal Article

Bulletin of the London Mathematical Society, ISSN 0024-6093, 04/2018, Volume 50, Issue 2, pp. 189 - 201

The Ramsey number rk(s,n) is the minimum N such that for every red–blue coloring of the k‐tuples of {1,…,N}, there are s integers such that every k‐tuple among...

05D40 | 05C65 (secondary) | 05D10 (primary) | MATHEMATICS

05D40 | 05C65 (secondary) | 05D10 (primary) | MATHEMATICS

Journal Article

Duke Mathematical Journal, ISSN 0012-7094, 11/2006, Volume 135, Issue 2, pp. 395 - 413

Let Q(n) denote a random symmetric (n x n)-matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables...

MATHEMATICS | 15A52 | 05D40

MATHEMATICS | 15A52 | 05D40

Journal Article

Proceedings of the London Mathematical Society, ISSN 0024-6115, 07/2019, Volume 119, Issue 1, pp. 55 - 103

In this paper, we give a lower bound for the maximum size of a k‐colored sum‐free set in Zmn, where k⩾3 and m⩾2 are fixed and n tends to infinity. If m is a...

05D40 | 11B30 (primary) | 05E15 (secondary) | MATHEMATICS | REMOVAL LEMMA

05D40 | 11B30 (primary) | 05E15 (secondary) | MATHEMATICS | REMOVAL LEMMA

Journal Article

Bulletin of the London Mathematical Society, ISSN 0024-6093, 04/2020, Volume 52, Issue 2, pp. 275 - 282

Pach showed that every d+1 sets of points Q1,…,Qd+1⊂Rd contain linearly sized subsets Pi⊂Qi such that all the transversal simplices that they span intersect....

52A35 (primary) | 05D99 | 05D40 | 52C10 (secondary) | MATHEMATICS | GRAPHS

52A35 (primary) | 05D99 | 05D40 | 52C10 (secondary) | MATHEMATICS | GRAPHS

Journal Article

Journal of the London Mathematical Society, ISSN 0024-6107, 12/2019, Volume 100, Issue 3, pp. 987 - 1012

We present a combinatorial approach to the infinitesimal distribution of the Gaussian orthogonal ensemble (goe). In particular, we show how the infinitesimal...

15B52 | 05D40 | 46L54 (primary) | 60B20 (secondary)

15B52 | 05D40 | 46L54 (primary) | 60B20 (secondary)

Journal Article

Journal of the London Mathematical Society, ISSN 0024-6107, 04/2018, Volume 97, Issue 2, pp. 247 - 257

We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4‐uniform Ramsey number...

05C65 | 05C55 | 05D10 | 05D40 (primary) | 05C15 | MATHEMATICS | NUMBERS | GRAPHS | Mathematics - Combinatorics

05C65 | 05C55 | 05D10 | 05D40 (primary) | 05C15 | MATHEMATICS | NUMBERS | GRAPHS | Mathematics - Combinatorics

Journal Article

Graphs and combinatorics, ISSN 1435-5914, 2012, Volume 29, Issue 1, pp. 1 - 38

The concept of rainbow connection was introduced by Chartrand et al. [14] in 2008. It is interesting and recently quite a lot papers have been published about...

68Q25 | (Strong) rainbow connection number | k -rainbow index | 05D40 | 05C40 | Mathematics | Engineering Design | Rainbow path | Algorithm | Computational complexity | Rainbow vertex-connection number | Rainbow k -connectivity | Combinatorics | 68R10 | 05C35 | 05C15 | k-rainbow index | Rainbow k-connectivity | NUMBERS | MINIMUM DEGREE | MATHEMATICS | DIAMETER | VERTEX-CONNECTION | COMPLEXITY | SUBGRAPHS | K-CONNECTIVITY | Surveys | Algorithms | Graph theory | Computational mathematics | Graph algorithms | Categories | Computation | Graphs | Rainbows | Joints | Combinatorial analysis | Complexity

68Q25 | (Strong) rainbow connection number | k -rainbow index | 05D40 | 05C40 | Mathematics | Engineering Design | Rainbow path | Algorithm | Computational complexity | Rainbow vertex-connection number | Rainbow k -connectivity | Combinatorics | 68R10 | 05C35 | 05C15 | k-rainbow index | Rainbow k-connectivity | NUMBERS | MINIMUM DEGREE | MATHEMATICS | DIAMETER | VERTEX-CONNECTION | COMPLEXITY | SUBGRAPHS | K-CONNECTIVITY | Surveys | Algorithms | Graph theory | Computational mathematics | Graph algorithms | Categories | Computation | Graphs | Rainbows | Joints | Combinatorial analysis | Complexity

Journal Article

Journal of Combinatorial Theory, Series B, ISSN 0095-8956, 01/2019, Volume 134, pp. 309 - 321

The r-uniform linear k-cycle Ckr is the r-uniform hypergraph on k(r−1) vertices whose edges are sets of r consecutive vertices in a cyclic ordering of the...

Balanced supersaturation | Asymptotic enumeration | MATHEMATICS | GRAPHS | Mathematics - Combinatorics

Balanced supersaturation | Asymptotic enumeration | MATHEMATICS | GRAPHS | Mathematics - Combinatorics

Journal Article

14.
Full Text
The Littlewood-Offord problem in high dimensions and a conjecture of Frankl and Füredi

Combinatorica, ISSN 0209-9683, 4/2012, Volume 32, Issue 3, pp. 363 - 372

We give a new bound on the probability that the random sum ξ 1 v 1+…+ξ n v n belongs to a ball of fixed radius, where the ξ i are i.i.d. Bernoulli random...

60C05 | Mathematics, general | Mathematics | Combinatorics | 05D40 | MATHEMATICS | LEMMA

60C05 | Mathematics, general | Mathematics | Combinatorics | 05D40 | MATHEMATICS | LEMMA

Journal Article

Combinatorica, ISSN 0209-9683, 2/2013, Volume 33, Issue 1, pp. 45 - 71

A Boolean function f: {0,1} n → {0,1} is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost...

60C05 | 06E30 | Mathematics, general | Mathematics | Combinatorics | 05D40 | MATHEMATICS | Computer science

60C05 | 06E30 | Mathematics, general | Mathematics | Combinatorics | 05D40 | MATHEMATICS | Computer science

Journal Article

Random Structures & Algorithms, ISSN 1042-9832, 01/2019, Volume 54, Issue 1, pp. 39 - 68

Write C(G) for the cycle space of a graph G, Cκ(G) for the subspace of C(G) spanned by the copies of the κ‐cycle Cκ in G, Tκ for the class of graphs satisfying...

sparse random graphs | cycle space | random graphs | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS | EIGENVALUES | MATHEMATICS, APPLIED | Graphs | Graph theory | Mathematics - Combinatorics

sparse random graphs | cycle space | random graphs | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS | EIGENVALUES | MATHEMATICS, APPLIED | Graphs | Graph theory | Mathematics - Combinatorics

Journal Article

Combinatorica, ISSN 0209-9683, 8/2017, Volume 37, Issue 4, pp. 617 - 632

In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed r∈ℕ, almost every n-vertex K r+1-free graph is r-partite. In this paper we extend this...

05C75 | 05C30 | 05D40 | Mathematics, general | Mathematics | Combinatorics | 05C35 | MATHEMATICS | ASYMPTOTIC NUMBER | SUBGRAPHS | HYPERGRAPHS | Resveratrol

05C75 | 05C30 | 05D40 | Mathematics, general | Mathematics | Combinatorics | 05C35 | MATHEMATICS | ASYMPTOTIC NUMBER | SUBGRAPHS | HYPERGRAPHS | Resveratrol

Journal Article

Combinatorica, ISSN 0209-9683, 8/2015, Volume 35, Issue 4, pp. 435 - 476

The first application of Szemerédi’s powerful regularity method was the following celebrated Ramsey-Turán result proved by Szemerédi in 1972: any K 4-free...

Mathematics, general | Mathematics | 05C55 | Combinatorics | 05C35 | 05D40 | MATHEMATICS | NUMBERS | BOUNDS | THEOREMS | PROOF | REGULARITY LEMMA | GRAPHS

Mathematics, general | Mathematics | 05C55 | Combinatorics | 05C35 | 05D40 | MATHEMATICS | NUMBERS | BOUNDS | THEOREMS | PROOF | REGULARITY LEMMA | GRAPHS

Journal Article

Combinatorica, ISSN 0209-9683, 8/2014, Volume 34, Issue 4, pp. 379 - 406

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs F and Г the generalized Turán density π F (Г) denotes the relative density of a...

Mathematics, general | Mathematics | 05C80 | Combinatorics | 05C35 | 05D40 | MATHEMATICS | SUBGRAPHS | TURANS THEOREM | Mathematics - Combinatorics

Mathematics, general | Mathematics | 05C80 | Combinatorics | 05C35 | 05D40 | MATHEMATICS | SUBGRAPHS | TURANS THEOREM | Mathematics - Combinatorics

Journal Article

Bulletin of the London Mathematical Society, ISSN 0024-6093, 08/2018, Volume 50, Issue 4, pp. 667 - 679

The influence of the kth coordinate on a Boolean function f:{0,1}n→{0,1} is the probability (with respect to the uniform measure on {0,1}n) that flipping xk...

06E30 | 05D40 (primary) | 05C35 (secondary) | MATHEMATICS | SENSITIVITY | THRESHOLDS | N-CUBE | INEQUALITY

06E30 | 05D40 (primary) | 05C35 (secondary) | MATHEMATICS | SENSITIVITY | THRESHOLDS | N-CUBE | INEQUALITY

Journal Article

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