The size‐Ramsey number of powers of paths

Maesaka

et al. 2020

J. London Math. Soc.

Self Cite

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.

“…The

k

th power

H^{k}

H

is the graph with vertex set

V (H)

in which there is an edge between distinct vertices

u

and

v

if and only if

u

and

v

are at distance at most

k

H

. 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.…”

Section: Introductionmentioning

confidence: 99%

The size‐Ramsey number of powers of bounded degree trees

Berger

Maesaka

et al. 2020

J. London Math. Soc.

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“…Given a graph H with n vertices and an integer k ě 2, the kth power of H, denoted H k , is the graph with vertex set V pHq where distinct vertices u and v are adjacent if and only if the distance between them in H is at most k. In [7], it was proved that powers of paths have linear 2-colour size-Ramsey numbers. Formally, Theorem 1.1 ([7]).…”

Section: §1 Introductionmentioning

confidence: 99%

The multicolour size-Ramsey number of powers of paths

Han

Jenssen

Journal of Combinatorial Theory, Series B

et al. 2020

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“…The following two lemmas are proved in [7]. Basically, together they imply that for every k and n there exists a graph G with O k (n) edges such that, for any disjoint sets of vertices A 1 , .…”

Section: Sparse Graphs With Many Long Pathsmentioning

confidence: 96%

“…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.…”

Section: Introductionmentioning

confidence: 99%

The size-Ramsey number of 3-uniform tight paths

Han

Letzter

et al. 2021

AiC

Self Cite

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).