The random <i>k</i>‐matching‐free process

Krivelevich, Michael; Kwan, Matthew; Loh, Po‐Shen; Sudakov, Benny

doi:10.1002/rsa.20814

Cited by 9 publications

(8 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These results were obtained independently by Glock, Kühn, Lo, and Osthus . Theorem For every

ℓ ⩾ 4

, there are

n_{ℓ}, β_{ℓ} > 0

such that, for all

n ⩾ n_{ℓ}

, there exists an

n

‐vertex partial Steiner triple system with at least

false(1 - n^{- β_{ℓ}} false) n^{2} / 6

triples and girth larger than

ℓ

.We prove Theorem by showing that the following natural (see ) constrained random process is very likely to produce the desired object for fixed

ℓ ⩾ 4

(see Theorem ). Beginning with the empty 3‐uniform hypergraph

H_{0}

n

vertices, we sequentially set

{scriptH}_{i + 1} : = {scriptH}_{i} + e_{i + 1}

, where the added triple

e_{i + 1}

is chosen uniformly at random from the collection of triples

x y z \notin {scriptH}_{i}

with the property that the girth of

{scriptH}_{i} + x y z

remains larger than

ℓ

(that is, that

{scriptH}_{i} + x y z

contains no set of

4 ⩽ a ⩽ ℓ

vertices that spans at least

a - 2

triples).…”

Section: Introductionmentioning

confidence: 86%

Large girth approximate Steiner triple systems

Bohman

Warnke

2019

J. London Math. Soc.

View full text Add to dashboard Cite

In 1973, Erdős asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g 4 for which some g-element vertex-set contains at least g − 2 triples.)We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed 4, we show that a natural constrained random process typically produces a partial Steiner triple system with (1/6 − o(1))n 2 triples and girth larger than . The process iteratively adds random triples subject to the constraint that the girth remains larger than . Our result is best possible up to the o(1)-term, which is a negative power of n.

show abstract

“…These results were obtained independently by Glock, Kühn, Lo, and Osthus . Theorem For every

ℓ ⩾ 4

, there are

n_{ℓ}, β_{ℓ} > 0

such that, for all

n ⩾ n_{ℓ}

, there exists an

n

‐vertex partial Steiner triple system with at least

false(1 - n^{- β_{ℓ}} false) n^{2} / 6

triples and girth larger than

ℓ

.We prove Theorem by showing that the following natural (see ) constrained random process is very likely to produce the desired object for fixed

ℓ ⩾ 4

(see Theorem ). Beginning with the empty 3‐uniform hypergraph

H_{0}

n

vertices, we sequentially set

{scriptH}_{i + 1} : = {scriptH}_{i} + e_{i + 1}

, where the added triple

e_{i + 1}

is chosen uniformly at random from the collection of triples

x y z \notin {scriptH}_{i}

with the property that the girth of

{scriptH}_{i} + x y z

remains larger than

ℓ

(that is, that

{scriptH}_{i} + x y z

contains no set of

4 ⩽ a ⩽ ℓ

vertices that spans at least

a - 2

triples).…”

Section: Introductionmentioning

confidence: 86%

Large girth approximate Steiner triple systems

Bohman

Warnke

2019

J. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…The question is of course again how long the process typically runs for. It was suggested by Krivelevich, Kwan, Loh, and Sudakov [22] that the process runs for quadratically many steps. We prove that with high probability, the number of leftover edges is o(n 2 ), implying Theorem 1.2.…”

Section: Theorem 13 ([24]mentioning

confidence: 99%

On a Conjecture of Erdős on Locally Sparse Steiner Triple Systems

Glock

Kühn

et al. 2020

Combinatorica

View full text Add to dashboard Cite

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.

show abstract

“…There are other types of random processes similar to the triangle-free process, which have striking applications in extremal graph theory. The H-free process was studied by Bohman and Keevash [5], and in a recent paper [6], the k-matching-free process was considered by Krivelevich, Kwan, Loh and Sudakov.…”

Section: Introductionmentioning

confidence: 99%

The Cyclic Triangle-Free Process

et al. 2019

View full text Add to dashboard Cite

For positive integers s and t, the Ramsey number R ( s , t ) is the smallest positive integer n such that every graph of order n contains either a clique of order s or an independent set of order t. The triangle-free process begins with an empty graph of order n, and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. It has been an important tool in studying the asymptotic lower bound for R ( 3 , t ) . Cyclic graphs are vertex-transitive. The symmetry of cyclic graphs makes it easier to compute their independent numbers than related general graphs. In this paper, the cyclic triangle-free process is studied. The sizes of the parameter sets and the independence numbers of the graphs obtained by the cyclic triangle-free process are studied. Lower bounds on R ( 3 , t ) for small t’s are computed, and R ( 3 , 35 ) ≥ 237 , R ( 3 , 36 ) ≥ 244 , R ( 3 , 37 ) ≥ 255 , R ( 3 , 38 ) ≥ 267 , etc. are obtained based on the graphs obtained by the cyclic triangle-free process. Finally, some problems on the cyclic triangle-free process and R ( 3 , t ) are proposed.

show abstract

The random k‐matching‐free process

Cited by 9 publications

References 40 publications

Large girth approximate Steiner triple systems

Large girth approximate Steiner triple systems

On a Conjecture of Erdős on Locally Sparse Steiner Triple Systems

The Cyclic Triangle-Free Process

Contact Info

Product

Resources

About