The multicolour size-Ramsey number of powers of paths

Han, Jie; Jenssen, Matthew; Kohayakawa, Yoshiharu; Mota, Guilherme Oliveira; Roberts, Barnaby

doi:10.1016/j.jctb.2020.06.004

Cited by 20 publications

(17 citation statements)

References 28 publications

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“…Recently it was proved that the 2‐colour size‐Ramsey number of powers of paths and cycles is linear [6]. This result was extended to any fixed number

s

of colours in [15], that is,

{\overset{r}{true}}_{s} false(P_{n}^{k} false) = O_{k, s} false(n false) and {\overset{r}{true}}_{s} false(C_{n}^{k} false) = O_{k, s} false(n false) .

In our main result (Theorem 1), we extend () to bounded powers of bounded degree trees. We prove that for any positive integers

k

and

s

, the

s

‐colour size‐Ramsey number of the

k

th power of any

n

‐vertex bounded degree tree is linear in

n

.…”

Section: Introductionmentioning

confidence: 78%

“…The proof of Theorem 1 follows the strategy developed in [15], proving the result by induction on the number of colours

s

. Very roughly speaking, we start with a graph

G

with suitable properties and, given any

s

‐colouring of the edges of

G

(

s ⩾ 2

), either we obtain a monochromatic copy of the power of the desired tree in

G

, or we obtain a large subgraph

H

G

that is coloured with at most

s - 1

colours; moreover, the graph

H

that we obtain is such that we can apply the induction hypothesis on it.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The size‐Ramsey number of powers of bounded degree trees

Berger

Kohayakawa

Maesaka

et al. 2020

J. London Math. Soc.

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Given a positive integer s, the s‐colour size‐Ramsey number of a graph H is the smallest integer m such that there exists a graph G with m edges with the property that, in any colouring of E(G) with s colours, there is a monochromatic copy of H. We prove that, for any positive integers k and s, the s‐colour size‐Ramsey number of the kth power of any n‐vertex bounded degree tree is linear in n. As a corollary, we obtain that the s‐colour size‐Ramsey number of n‐vertex graphs with bounded treewidth and bounded degree is linear in n, which answers a question raised by Kamčev, Liebenau, Wood and Yepremyan.

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“…Recently it was proved that the 2‐colour size‐Ramsey number of powers of paths and cycles is linear [6]. This result was extended to any fixed number

s

of colours in [15], that is,

{\overset{r}{true}}_{s} false(P_{n}^{k} false) = O_{k, s} false(n false) and {\overset{r}{true}}_{s} false(C_{n}^{k} false) = O_{k, s} false(n false) .

In our main result (Theorem 1), we extend () to bounded powers of bounded degree trees. We prove that for any positive integers

k

and

s

, the

s

‐colour size‐Ramsey number of the

k

th power of any

n

‐vertex bounded degree tree is linear in

n

.…”

Section: Introductionmentioning

confidence: 78%

“…The proof of Theorem 1 follows the strategy developed in [15], proving the result by induction on the number of colours

s

. Very roughly speaking, we start with a graph

G

with suitable properties and, given any

s

‐colouring of the edges of

G

(

s ⩾ 2

), either we obtain a monochromatic copy of the power of the desired tree in

G

, or we obtain a large subgraph

H

G

that is coloured with at most

s - 1

colours; moreover, the graph

H

that we obtain is such that we can apply the induction hypothesis on it.…”

Section: Introductionmentioning

confidence: 99%

The size‐Ramsey number of powers of bounded degree trees

Berger

Kohayakawa

Maesaka

et al. 2020

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Our proof combines new ideas and the method developed by Clemens, Jenssen, Kohayakawa, Morrison, Mota, Reding, and Roberts [7] for estimating the size-Ramsey number of powers of paths (see also [5,12]). It is plausible that ideas from [12,5] may provide a strategy to solve the case with s ≥ 3 colours. However, the question whether the size-Ramsey number of a tight path is linear for hypergraphs with uniformity r ≥ 4 remains open and requires additional ideas.…”

Section: Introductionmentioning

confidence: 99%

The size-Ramsey number of 3-uniform tight paths

Han

Kohayakawa

Letzter

et al. 2021

AiC

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Given a hypergraph H, the size-Ramsey number r(H) is the smallest integer m such that there exists a graph G with m edges with the property that in any colouring of the edges of G with two colours there is amonochromatic copy of H. We prove that the size-Ramsey number of the 3-uniform tight path on n vertices P_n is linear in n, i.e., r(P_n)=O(n). This answers a question by Dudek, Fleur, Mubayi, and Rödl for 3-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved r(P_n)=O(n^1.5*log^1.5 n).

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“…Ramsey theory [1] has played an important branch in combinatorics, which spans numerous diverse areas of mathematics. Many research efforts have been devoted into computing Ramsey numbers and their generalizations [2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

Random Cyclic Triangle-Free Graphs of Prime Order

Jiang

Liang

Teng

et al. 2021

Journal of Mathematics

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Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process. It begins with an empty graph of order n and generates a cyclic graph of order n by iteratively adding parameters, chosen uniformly at random, subject to the constraint that no triangle is formed in the cyclic graph obtained, until no more parameters can be added. The structure of a cyclic triangle-free graph of the prime order is different from that of composite integer order. Cyclic graphs of prime order have better properties than those of composite number order, which enables generating cyclic triangle-free graphs more efficiently. In this paper, a novel approach to generating cyclic triangle-free graphs of prime order is proposed. Based on the cyclic graphs of prime order, obtained by the CTFP and its variant, many new lower bounds on R 3 , t are computed, including R 3,34 ≥ 230 , R 3,35 ≥ 242 , R 3,36 ≥ 252 , R 3,37 ≥ 264 , R 3,38 ≥ 272 . Our experimental results demonstrate that all those related best known lower bounds, except the bound on R 3,34 , are improved by 5 or more.

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The multicolour size-Ramsey number of powers of paths

Cited by 20 publications

References 28 publications

The size‐Ramsey number of powers of bounded degree trees

The size‐Ramsey number of powers of bounded degree trees

The size-Ramsey number of 3-uniform tight paths

Random Cyclic Triangle-Free Graphs of Prime Order

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