A note on the random greedy independent set algorithm

Bennett, Patrick J.; Bohman, Tom

doi:10.1002/rsa.20667

Cited by 29 publications

(80 citation statements)

References 34 publications

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“…We remark that in [40] the authors stated that in a private communication, Jacques Verstraëte suggested that a lower bound on ex r (n, B t ), which is exactly the same with Proposition 8, can also be proved by using the method of [8,7] (which is rather involved). Nevertheless, since [40] stated this result (as well as Proposition 13 below) without a proof, we present it here as an easy consequence of Theorem 3.…”

Section: R-graphs With No Short Berge Cyclesmentioning

confidence: 59%

Universally Sparse Hypergraphs with Applications to Coding Theory

Shangguan

Tamo

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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For fixed integers r ≥ 3, e ≥ 3, v ≥ r + 1, an r-uniform hypergraph is called G r (v, e)-free if the union of any e distinct edges contains at least v + 1 vertices. Brown, Erdős and Sós showed that the maximum number of edges of such a hypergraph on n vertices, denoted as f r (n, v, e), satisfiesFor e − 1 | er − v, the lower bound matches the upper bound up to a constant factor; whereas for e − 1 ∤ er − v, in general it is a notoriously hard problem to determine the correct exponent of n.Among other results, we improve the above lower bound by showing thatfor any r, e, v satisfying gcd(e − 1, er − v) = 1. The hypergraph we constructed is in fact G r (ir − ⌈ (i−1)(er−v) e−1 ⌉, i)-free for every 2 ≤ i ≤ e, and it has several interesting applications in Coding Theory. The proof of the new lower bound is based on a novel application of the lower bound on the hypergraph independence number due to Duke, Lefmann, and Rödl.

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Section: R-graphs With No Short Berge Cyclesmentioning

confidence: 59%

Universally Sparse Hypergraphs with Applications to Coding Theory

Shangguan

Tamo

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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“…holds for all n. This was first proved by Bohman and Keevash [11] and later generalized to hypergraphs of higher uniformity by Bennett and Bohman [10].…”

Section: Proofs Of Theorems 2 Andmentioning

confidence: 76%

Supersaturated Sparse Graphs and Hypergraphs

Ferber

McKinley

Samotij

2018

International Mathematics Research Notices

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A central problem in extremal graph theory is to estimate, for a given graph H, the number of H-free graphs on a given set of n vertices. In the case when H is not bipartite, fairly precise estimates on this number are known. In particular, thirty years ago, Erdős, Frankl, and Rödl proved that there are 2 (1+o(1))ex(n,H) such graphs. In the bipartite case, however, nontrivial bounds have been proven only for relatively few special graphs H.We make a first attempt at addressing this enumeration problem for a general bipartite graph H. We show that an upper bound of 2 O(ex(n,H)) on the number of H-free graphs with n vertices follows merely from a rather natural assumption on the growth rate of n → ex(n, H); an analogous statement remains true when H is a uniform hypergraph. Subsequently, we derive several new results, along with most previously known estimates, as simple corollaries of our theorem. At the heart of our proof lies a general supersaturation statement that extends the seminal work of Erdős and Simonovits. The bounds on the number of H-free hypergraphs are derived from it using the method of hypergraph containers.

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“…For a fixed (hyper-)graph H, the H-free process has been extensively studied, in particular if H is 'strictly 2-balanced' (see e.g. [2,4,21,27,32,33]). A particular challenge arising in the analysis of the current process is that each individual Erdős-configuration in H has a significant influence on the trajectory of the process.…”

Section: Theorem 13 ([24]mentioning

confidence: 99%

“…The following is an immediate consequence of Proposition 5.9. For distinct triples T 1 , T 2 , we let X T 1 ,T 2 (i) be the set of all pairs S 1 = S 2 , not both diamonds, such that for each ℓ ∈ [2], S ℓ ∈ X T ℓ ,j ℓ ,j ℓ −4 (i) with 4 ≤ j ℓ ≤ j max , and such that (…”

Section: Counting Double Configurationsmentioning

confidence: 99%

On a Conjecture of Erdős on Locally Sparse Steiner Triple Systems

Glock

Kühn

et al. 2020

Combinatorica

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A famous theorem of Kirkman says that there exists a Steiner triple system of order n if and only if n ≡ 1, 3 mod 6. In 1973, Erdős conjectured that one can find so-called 'sparse' Steiner triple systems. Roughly speaking, the aim is to have at most j − 3 triples on every set of j points, which would be best possible. (Triple systems with this sparseness property are also referred to as having high girth.) We prove this conjecture asymptotically by analysing a natural generalization of the triangle removal process. Our result also solves a problem posed by Lefmann, Phelps and Rödl as well as Ellis and Linial in a strong form, and answers a question of Krivelevich, Kwan, Loh, and Sudakov. Moreover, we pose a conjecture which would generalize the Erdős conjecture to Steiner systems with arbitrary parameters and provide some evidence for this.

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A note on the random greedy independent set algorithm

Cited by 29 publications

References 34 publications

Universally Sparse Hypergraphs with Applications to Coding Theory

Universally Sparse Hypergraphs with Applications to Coding Theory

Supersaturated Sparse Graphs and Hypergraphs

On a Conjecture of Erdős on Locally Sparse Steiner Triple Systems

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