scholarly journals The size-Ramsey number of 3-uniform tight paths

2021 ◽  
Author(s):  
Jie Han ◽  
Yoshiharu Kohayakawa ◽  
Shoham Letzter ◽  
Guilherme Oliveira Mota ◽  
Olaf Parczyk
Keyword(s):  
Graph Theory ◽  
Ramsey Number ◽  
Uniform Hypergraphs

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).

scholarly journals A Note on Embedding Hypertrees

10.37236/256 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Po-Shen Loh
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Classical Result ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs

A classical result from graph theory is that every graph with chromatic number $\chi > t$ contains a subgraph with all degrees at least $t$, and therefore contains a copy of every $t$-edge tree. Bohman, Frieze, and Mubayi recently posed this problem for $r$-uniform hypergraphs. An $r$-tree is a connected $r$-uniform hypergraph with no pair of edges intersecting in more than one vertex, and no sequence of distinct vertices and edges $(v_1, e_1, \ldots, v_k, e_k)$ with all $e_i \ni \{v_i, v_{i+1}\}$, where we take $v_{k+1}$ to be $v_1$. Bohman, Frieze, and Mubayi proved that $\chi > 2rt$ is sufficient to embed every $r$-tree with $t$ edges, and asked whether the dependence on $r$ was necessary. In this note, we completely solve their problem, proving the tight result that $\chi > t$ is sufficient to embed any $r$-tree with $t$ edges.

scholarly journals Anti-Ramsey number of matchings in r-partite r-uniform hypergraphs

2022 ◽  
Vol 345 (4) ◽  
pp. 112782
Author(s):  
Yisai Xue ◽  
Erfang Shan ◽  
Liying Kang
Keyword(s):  
Ramsey Number ◽  
Uniform Hypergraphs

scholarly journals Equitable Hypergraph Orientations

10.37236/608 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Yair Caro ◽  
Douglas West ◽  
Raphael Yuster
Keyword(s):  
Graph Theory ◽  
Classical Result ◽  
Uniform Hypergraphs

A classical result in graph theory asserts that every graph can be oriented so that the indegree and outdegree of each vertex differ by at most $1$. We study the extent to which the result generalizes to uniform hypergraphs.

scholarly journals The Ramsey Number of Loose Paths in 3-Uniform Hypergraphs

10.37236/2725 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Leila Maherani ◽  
Gholam Reza Omidi ◽  
Ghaffar Raeisi ◽  
Maryam Shahsiah
Keyword(s):  
The Other ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Uniform Hypergraphs ◽  
Asymptotic Values

Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of $3$-uniform loose paths when one of the paths is significantly larger than the other:  for every $n\geq \Big\lfloor\frac{5m}{4}\Big\rfloor$, we show that $$R(\mathcal{P}^3_n,\mathcal{P}^3_m)=2n+\Big\lfloor\frac{m+1}{2}\Big\rfloor.$$

scholarly journals Cycles Are Strongly Ramsey-Unsaturated

2014 ◽  
Vol 23 (4) ◽  
pp. 607-630 ◽  
Author(s):  
J. SKOKAN ◽  
M. STEIN
Keyword(s):  
Graph Theory ◽  
Maximum Degree ◽  
Ramsey Number ◽  
Odd Cycle

We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp in [J. Graph Theory51 (2006), pp. 22–32], where it is shown that cycles (except for C4) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle Cn, unless n is even and adding the chord creates an odd cycle.We prove this conjecture for large cycles by showing a stronger statement. If a graph H is obtained by adding a linear number of chords to a cycle Cn, then r(H)=r(Cn), as long as the maximum degree of H is bounded, H is either bipartite (for even n) or almost bipartite (for odd n), and n is large.This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses the regularity method.

The 3-Colour Ramsey Number of a 3-Uniform Berge Cycle

2010 ◽  
Vol 20 (1) ◽  
pp. 53-71 ◽  
Author(s):  
ANDRÁS GYÁRFÁS ◽  
GÁBOR N. SÁRKÖZY
Keyword(s):  
Ramsey Number ◽  
Regularity Lemma ◽  
Ramsey Numbers ◽  
Uniform Hypergraphs

The asymptotics of 2-colour Ramsey numbers of loose and tight cycles in 3-uniform hypergraphs were recently determined [16, 17]. We address the same problem for Berge cycles and for 3 colours. Our main result is that the 3-colour Ramsey number of a 3-uniform Berge cycle of length n is asymptotic to $\frac{5n}{4}$. The result is proved with the Regularity Lemma via the existence of a monochromatic connected matching covering asymptotically 4n/5 vertices in the multicoloured 2-shadow graph induced by the colouring of Kn(3).

Tripartite and Quadpartite Size Ramsey Numbers for All Pairs of Connected Graphs on Four Vertices

2020 ◽  
pp. 251-266
Author(s):  
Chula J. Jayawardene
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Ramsey Theory ◽  
Premature Death ◽  
Formal Logic ◽  
Young Age ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Connected Graphs ◽  
Area Of Study

A popular area of graph theory is based on a paper written in 1930 by F. P. Ramsey titled “On a Problem on Formal Logic.” A theorem which was proved in his paper triggered the study of modern Ramsey theory. However, his premature death at the young age of 26 hindered the development of this area of study at the initial stages. The balanced size multipartite Ramsey number mj (H,G) is defined as the smallest positive number s such that Kj×s→ (H,G). There are 36 pairs of (H, G), when H, G represent connected graphs on four vertices (as there are only 6 non-isomorphic connected graphs on four vertices). In this chapter, the authors find mj (H, G) exhaustively for all such pairs in the tripartite case j=3, and in the quadpartite case j=4, excluding the case m4 (K4,K4). In this case, the only known result is that m4 (K4,K4) is greater than or equal to 4, since no upper bound has been found as yet.

scholarly journals A note on planar Ramsey numbers for a triangle versus wheels

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Guofei Zhou ◽  
Yaojun Chen ◽  
Zhengke Miao ◽  
Shariefuddin Pirzada
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
International Audience

Graph Theory International audience For two given graphs G and H , the planar Ramsey number P R ( G; H ) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G , or its complement contains a copy of H . In this paper, we determine all planar Ramsey numbers for a triangle versus wheels.

scholarly journals The Ramsey Number of Fano Plane Versus Tight Path

10.37236/8374 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
József Balogh ◽  
Felix Christian Clemen ◽  
Jozef Skokan ◽  
Adam Zsolt Wagner
Keyword(s):  
Edge Coloring ◽  
Ramsey Number ◽  
Short Paper ◽  
Simple Construction ◽  
Uniform Hypergraph ◽  
Fano Plane ◽  
Blue Edge ◽  
Uniform Hypergraphs ◽  
Plane Versus

The hypergraph Ramsey number of two $3$-uniform hypergraphs $G$ and $H$, denoted by $R(G,H)$, is the least integer~$N$ such that every red-blue edge-coloring of the complete $3$-uniform hypergraph on $N$ vertices contains a red copy of $G$ or a blue copy of $H$. The Fano plane $\mathbb{F}$ is the unique 3-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. There is a simple construction showing that $R(H,\mathbb{F}) \ge 2(v(H)-1) + 1.$  Hypergraphs $H$ for which the equality holds are called $\mathbb{F}$-good. Conlon asked to determine all $H$ that are $\mathbb{F}$-good.In this short paper we make progress on this problem and prove that the tight path of length $n$ is $\mathbb{F}$-good.

Sign in / Sign up

Export Citation Format

Share Document