The size Ramsey number of a directed path

Ben-Eliezer, Ido; Krivelevich, Michael; Sudakov, Benny

doi:10.1016/j.jctb.2011.10.002

Cited by 69 publications

(95 citation statements)

References 17 publications

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“…To prove Lemma , we analyze a depth‐first search algorithm, adapting a proof idea in , Lemma 4.4]. More specifically, we run an algorithm (stated formally as Algorithm ).…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…This was negatively answered by Rödl and Szemerédi, who showed that there exists an

n

‐vertex graph

H

and maximum degree

3

such that

\hat{r} (H) = Ω (n \log^{1 ∕ 60} n)

. The current best upper bound for bounded‐degree graphs is proved in , where it is shown that for every

normalΔ

there is a constant

c

such that for any graph

H

with

n

vertices and maximum degree

normalΔ

\hat{r} (H) \leq c n^{2 - 1 ∕ Δ} \log^{1 ∕ Δ} n .

For further results on size‐Ramsey numbers, the reader is referred to .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The size‐Ramsey number of powers of paths

Clemens

Jenssen

Kohayakawa

et al. 2018

Journal of Graph Theory

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Given graphs G and H and a positive integer q, say that G is q‐Ramsey for H, denoted G→(H)q, if every q‐coloring of the edges of G contains a monochromatic copy of H. The size‐Ramsey number truerˆ(H) of a graph H is defined to be truerˆ(H)=min{∣E(G)∣:G→(H)2}. Answering a question of Conlon, we prove that, for every fixed k, we have truerˆ(Pnk)=O(n), where Pnk is the kth power of the n‐vertex path Pn (ie, the graph with vertex set V(Pn) and all edges {u,v} such that the distance between u and v in Pn is at most k). Our proof is probabilistic, but can also be made constructive.

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“…To prove Lemma , we analyze a depth‐first search algorithm, adapting a proof idea in , Lemma 4.4]. More specifically, we run an algorithm (stated formally as Algorithm ).…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…This was negatively answered by Rödl and Szemerédi, who showed that there exists an

n

‐vertex graph

H

and maximum degree

3

such that

\hat{r} (H) = Ω (n \log^{1 ∕ 60} n)

. The current best upper bound for bounded‐degree graphs is proved in , where it is shown that for every

normalΔ

there is a constant

c

such that for any graph

H

with

n

vertices and maximum degree

normalΔ

\hat{r} (H) \leq c n^{2 - 1 ∕ Δ} \log^{1 ∕ Δ} n .

For further results on size‐Ramsey numbers, the reader is referred to .…”

Section: Introductionmentioning

confidence: 99%

The size‐Ramsey number of powers of paths

Clemens

Jenssen

Kohayakawa

et al. 2018

Journal of Graph Theory

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show abstract

“…We next show that a bipartite directed graph with a simple expansion property contains a long directed path. The proof follows ideas from .…”

Section: The Regularity Lemma For Sparse Directed Graphs and Long Patmentioning

confidence: 99%

Long cycles in subgraphs of (pseudo)random directed graphs

Ben-Eliezer

Krivelevich

Sudakov

2011

Journal of Graph Theory

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We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 0 < γ < 1/2 we find a constant c = c(γ) such that the following holds. Let G = (V, E) be a (pseudo)random directed graph on n vertices, and let G ′ be a subgraph of G with (1/2 + γ)|E| edges. Then G ′ contains a directed cycle of length at least (c − o(1))n. Moreover, there is a subgraph G ′′ of G with (1/2 + γ − o(1))|E| edges that does not contain a cycle of length at least cn.

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“…This question was implicitly posed by Ben‐Eliezer, Krivelevich, and Sudakov in , where they note that

m (T) \leq \frac{2 n}{\sqrt{\log n}}

for any n ‐vertex tournament. Indeed, it is well known and easy to see that any tournament of order n has a transitive subtournament of order

\log n

.…”

Section: Introductionmentioning

confidence: 99%

“…Here we consider the following natural analogue for oriented paths. Given an oriented graph H , we define the oriented size Ramsey number , denoted by