scholarly journals Ramsey Numbers of Berge-Hypergraphs and Related Structures

10.37236/8892 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Nika Salia ◽  
Casey Tompkins ◽  
Zhiyu Wang ◽  
Oscar Zamora
Keyword(s):  
Edge Coloring ◽  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Ramsey Numbers

For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists an injection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be the smallest integer $n$ such that for any $2$-edge-coloring of a complete $r$-uniform hypergraph on $n$ vertices, there is a monochromatic Berge-$G$ subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that $R^3(BK_s, BK_t) = s+t-3$ for $s,t \geq 4$ and $\max(s,t) \geq 5$ where $BK_n$ is a Berge-$K_n$ hypergraph. For higher uniformity, we show that $R^4(BK_t, BK_t) = t+1$ for $t\geq 6$ and $R^k(BK_t, BK_t)=t$ for $k \geq 5$ and $t$ sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.

scholarly journals Ramsey numbers for certain k-graphs

1981 ◽  
Vol 30 (3) ◽  
pp. 284-296 ◽  
Author(s):  
Stefan A. Burr ◽  
Richard A. Duke
Keyword(s):  
Lower Bounds ◽  
Complete Solution ◽  
Ramsey Number ◽  
Upper And Lower Bounds ◽  
Uniform Hypergraph ◽  
Steiner Systems ◽  
Ramsey Numbers ◽  
Existence Question

AbstractWe are interested here in the Ramsey number r(T, C), where C is a complete k-uniform hypergraph and T is a “tree-like” k-graph. Upper and lower bounds are found for these numbers which lead, in some cases, to the exact value for r(T, C) and to a generalization of a theorem of Chváta1 on Ramsey numbers for graphs. In other cases we show that a determination of the exact values of r(T, C) would be equivalent to obtaining a complete solution to existence question for a certain class of Steiner systems.

scholarly journals Anti-Ramsey Numbers in Complete k-Partite Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Jili Ding ◽  
Hong Bian ◽  
Haizheng Yu
Keyword(s):  
Spanning Trees ◽  
Edge Coloring ◽  
Perfect Matchings ◽  
Ramsey Number ◽  
Hamilton Cycles ◽  
Ramsey Numbers ◽  
The Family

The anti-Ramsey number ARG,H is the maximum number of colors in an edge-coloring of G such that G contains no rainbow subgraphs isomorphic to H. In this paper, we discuss the anti-Ramsey numbers ARKp1,p2,…,pk,Tn, ARKp1,p2,…,pk,ℳ, and ARKp1,p2,…,pk,C of Kp1,p2,…,pk, where Tn,ℳ, and C denote the family of all spanning trees, the family of all perfect matchings, and the family of all Hamilton cycles in Kp1,p2,…,pk, respectively.

scholarly journals Degree Ramsey Numbers of Closed Blowups of Trees

10.37236/2526 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Paul Horn ◽  
Kevin G. Milans ◽  
Vojtěch Rödl
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Edge Coloring ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Delta G ◽  
Monochromatic Copy

The degree Ramsey number of a graph $G$, denoted $R_\Delta(G;s)$, is $\min\{\Delta(H)\colon\, H\stackrel{s}{\to} G\}$, where $H\stackrel{s}{\to} G$ means that every $s$-edge-coloring of $H$ contains a monochromatic copy of $G$.  The closed $k$-blowup of a graph is obtained by replacing every vertex with a clique of size $k$ and every edge with a complete bipartite graph where both partite sets have size $k$.  We prove that there is a function $f$ such that $R_\Delta(G;s) \le f(\Delta(G), s)$  when $G$ is a closed blowup of a tree.

scholarly journals Off-Diagonal Ordered Ramsey Numbers of Matchings

10.37236/8085 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Dhruv Rohatgi
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Edge Coloring ◽  
Ramsey Number ◽  
Relative Order ◽  
Ramsey Numbers ◽  
Asymptotic Bounds ◽  
Special Cases ◽  
Ordered Graphs

For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete ordered graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the off-diagonal ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the maximum is over all ordered matchings $M$ on $n$ vertices. The best-known upper bound is $O(n^2/\log n)$, whereas the best-known lower bound is $\Omega((n/\log n)^{4/3})$, and Conlon et al. hypothesize that there is some fixed $\epsilon > 0$ such that $r_<(M, K_3) = O(n^{2-\epsilon})$ for every ordered matching $M$. We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for ordered matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random ordered matchings with interval chromatic number $2$.

scholarly journals Anti-Ramsey Numbers for Graphs with Independent Cycles

10.37236/174 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Zemin Jin ◽  
Xueliang Li
Keyword(s):  
Edge Coloring ◽  
Ramsey Number ◽  
Colored Graph ◽  
Ramsey Numbers ◽  
The Family

An edge-colored graph is called rainbow if all the colors on its edges are distinct. Let ${\cal G}$ be a family of graphs. The anti-Ramsey number $AR(n,{\cal G})$ for ${\cal G}$, introduced by Erdős et al., is the maximum number of colors in an edge coloring of $K_n$ that has no rainbow copy of any graph in ${\cal G}$. In this paper, we determine the anti-Ramsey number $AR(n,\Omega_2)$, where $\Omega_2$ denotes the family of graphs that contain two independent cycles.

scholarly journals Multicolor Ramsey Numbers and Restricted Turán Numbers for the Loose 3-Uniform Path of Length Three

10.37236/7119 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Andrzej Ruciński ◽  
Eliza Jackowska ◽  
Joanna Polcyn
Keyword(s):  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Ramsey Numbers ◽  
Turán Number ◽  
Standard Application

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$ for all $r\ge3$. The latter result follows by a standard application of the Turán number $\mathrm{ex}_3(n;P)$, which was determined to be $\binom{n-1}2$ in our previous work. We have also shown that the full star is the only extremal 3-graph for $P$. In this paper, we perform a subtle analysis of the Turán numbers for $P$ under some additional restrictions. Most importantly, we determine the largest number of edges in an $n$-vertex $P$-free 3-graph which is not a star. These Turán-type results, in turn, allow us to confirm the formula $R(P;r)=r+6$ for $r\in\{4,5,6,7\}$.

Anti-Ramsey numbers for matchings in regular bipartite graphs

2017 ◽  
Vol 09 (02) ◽  
pp. 1750019 ◽  
Author(s):  
Zemin Jin ◽  
Oothan Nweit ◽  
Kaijun Wang ◽  
Yuling Wang
Keyword(s):  
Edge Coloring ◽  
Bipartite Graphs ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Number Formula

Let [Formula: see text] be a family of graphs. The anti-Ramsey number [Formula: see text] for [Formula: see text] in the graph [Formula: see text] is the maximum number of colors in an edge coloring of [Formula: see text] that does not have any rainbow copy of any graph in [Formula: see text]. In this paper, we consider the anti-Ramsey number for matchings in regular bipartite graphs and determine its value under several conditions.

scholarly journals On Ramsey numbers of hedgehogs

2019 ◽  
Vol 29 (1) ◽  
pp. 101-112
Author(s):  
Jacob Fox ◽  
Ray Li
Keyword(s):  
Ramsey Number ◽  
Square Root ◽  
Uniform Hypergraph ◽  
Ramsey Numbers ◽  
Unique Vertex

AbstractThe hedgehog Ht is a 3-uniform hypergraph on vertices $1, \ldots ,t + \left({\matrix{t \cr 2}}\right)$ such that, for any pair (i, j) with 1 ≤ i < j ≤ t, there exists a unique vertex k > t such that {i, j, k} is an edge. Conlon, Fox and Rödl proved that the two-colour Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-colour Ramsey number grows exponentially in the square root of the number of vertices. They asked whether the two-colour Ramsey number of the hedgehog Ht is nearly linear in the number of its vertices. We answer this question affirmatively, proving that r(Ht) = O(t2 ln t).

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.

scholarly journals A Note on On-Line Ramsey Numbers for Some Paths

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 735
Author(s):  
Tomasz Dzido ◽  
Renata Zakrzewska
Keyword(s):  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
On Line ◽  
Infinite Set ◽  
Do So

We consider the important generalisation of Ramsey numbers, namely on-line Ramsey numbers. It is easiest to understand them by considering a game between two players, a Builder and Painter, on an infinite set of vertices. In each round, the Builder joins two non-adjacent vertices with an edge, and the Painter colors the edge red or blue. An on-line Ramsey number r˜(G,H) is the minimum number of rounds it takes the Builder to force the Painter to create a red copy of graph G or a blue copy of graph H, assuming that both the Builder and Painter play perfectly. The Painter’s goal is to resist to do so for as long as possible. In this paper, we consider the case where G is a path P4 and H is a path P10 or P11.

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