scholarly journals On the Multicolor Ramsey Number for 3-Paths of Length Three

10.37236/6670 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Tomasz Łuczak ◽  
Joanna Polcyn
Keyword(s):  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Multicolor Ramsey Number

We show that if we color the hyperedges of the complete $3$-uniform hypergraph on $2n+\sqrt{18n+1}+2$ vertices with $n$ colors, then one of the color classes contains a loose path of length three.

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 On the multicolor Ramsey number of a graph with m edges

2016 ◽  
Vol 339 (12) ◽  
pp. 2857-2860 ◽  
Author(s):  
Kathleen Johst ◽  
Yury Person
Keyword(s):  
Ramsey Number ◽  
Multicolor Ramsey Number

scholarly journals On Berge-Ramsey Problems

10.37236/8775 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Dániel Gerbner
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Constant Term ◽  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Large Margin

Given a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ if $V(G)\subset V(\mathcal{H})$ and there is a bijection $f:E(G)\rightarrow E(\mathcal{H})$ such that for any edge $e$ of $G$ we have $e\subset f(e)$. We study Ramsey problems for Berge copies of graphs, i.e. the smallest number of vertices of a complete $r$-uniform hypergraph, such that if we color the hyperedges with $c$ colors, there is a monochromatic Berge copy of $G$. We obtain a couple results regarding these problems. In particular, we determine for which $r$ and $c$ the Ramsey number can be super-linear. We also show a new way to obtain lower bounds, and improve the general lower bounds by a large margin. In the specific case $G=K_n$ and $r=2c-1$, we obtain an upper bound that is sharp besides a constant term, improving earlier results.

scholarly journals On Some Ramsey and Turán-Type Numbers for Paths and Cycles

10.37236/1081 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Tomasz Dzido ◽  
Marek Kubale ◽  
Konrad Piwakowski
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Multicolor Ramsey Number ◽  
Monochromatic Copy

For given graphs $G_{1}, G_{2}, ... , G_{k}$, where $k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, ... , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph on $n$ vertices with $k$ colors, there is always a monochromatic copy of $G_{i}$ colored with $i$, for some $1 \leq i \leq k$. Let $P_k$ (resp. $C_k$) be the path (resp. cycle) on $k$ vertices. In the paper we show that $R(P_3,C_k,C_k)=R(C_k,C_k)=2k-1$ for odd $k$. In addition, we provide the exact values for Ramsey numbers $R(P_{4}, P_{4}, C_{k})=k+2$ and $R(P_{3}, P_{5}, C_{k})=k+1$.

scholarly journals New upper bound for multicolor Ramsey number of odd cycles

2019 ◽  
Vol 342 (1) ◽  
pp. 217-220 ◽  
Author(s):  
Qizhong Lin ◽  
Weiji Chen
Keyword(s):  
Upper Bound ◽  
Ramsey Number ◽  
Multicolor Ramsey Number ◽  
Odd Cycles

On the Erdős–Ginzburg–Ziv invariant and zero-sum Ramsey number for intersecting families

2014 ◽  
Vol 10 (07) ◽  
pp. 1637-1647 ◽  
Author(s):  
Haiyan Zhang ◽  
Guoqing Wang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Zero Sum

Let G be a finite abelian group, and let m > 0 with exp (G) | m. Let sm(G) be the generalized Erdős–Ginzburg–Ziv invariant which denotes the smallest positive integer d such that any sequence of elements in G of length d contains a subsequence of length m with sum zero in G. For any integer r > 0, let [Formula: see text] be the collection of all r-uniform intersecting families of size m. Let [Formula: see text] be the smallest positive integer d such that any G-coloring of the edges of the complete r-uniform hypergraph [Formula: see text] yields a zero-sum copy of some intersecting family in [Formula: see text]. Among other results, we mainly prove that [Formula: see text], where Ω(sm(G)) denotes the least positive integer n such that [Formula: see text], and we show that if r | Ω(sm(G)) – 1 then [Formula: see text].

scholarly journals On the Weight of Berge-$F$-Free Hypergraphs

10.37236/8504 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Sean English ◽  
Dániel Gerbner ◽  
Abhishek Methuku ◽  
Cory Palmer
Keyword(s):  
Short Note ◽  
Ramsey Number ◽  
Weight Functions ◽  
Uniform Hypergraph ◽  
The Way

For a graph $F$, we say a hypergraph is a Berge-$F$ if it can be obtained from $F$ by replacing each edge of $F$ with a hyperedge containing it. A hypergraph is Berge-$F$-free if it does not contain a subhypergraph that is a Berge-$F$. The weight of a non-uniform hypergraph $\mathcal{H}$ is the quantity $\sum_{h \in E(\mathcal{H})} |h|$. Suppose $\mathcal{H}$ is a Berge-$F$-free hypergraph on $n$ vertices. In this short note, we prove that as long as every edge of $\mathcal{H}$ has size at least the Ramsey number of $F$, the weight of $\mathcal{H}$ is $o(n^2)$. This result is best possible in some sense. Along the way, we study other weight functions, and strengthen results of Gerbner and Palmer; and Grósz, Methuku and Tompkins.

scholarly journals On Multicolor Ramsey Number of Paths Versus Cycles

10.37236/511 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Gholam Reza Omidi ◽  
Ghaffar Raeisi
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Complete Graph ◽  
Ramsey Number ◽  
Subgraph Isomorphic ◽  
Multicolor Ramsey Number

Let $G_1, G_2, G_3, \ldots , G_t$ be graphs. The multicolor Ramsey number $R(G_1, G_2, \ldots, G_t)$ is the smallest positive integer $n$ such that if the edges of a complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs $H_1,H_2,\ldots,H_t$, then at least one $H_i$ has a subgraph isomorphic to $G_i$. In this paper, we provide the exact value of $R(P_{n_1}, P_{n_2},\ldots, P_{n_t},C_k)$ for certain values of $n_i$ and $k$. In addition, the exact values of $R(P_5,C_4,P_k)$, $R(P_4,C_4,P_k)$, $R(P_5,P_5,P_k)$ and $R(P_5,P_6,P_k)$ are given. Finally, we give a lower bound for $R(P_{2n_1}, P_{2n_2},\ldots, P_{2n_t})$ and we conjecture that this lower bound is the exact value of this number. Moreover, some evidence is given for this conjecture.

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.

The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

2009 ◽  
Vol 18 (1-2) ◽  
pp. 165-203 ◽  
Author(s):  
P. E. HAXELL ◽  
T. ŁUCZAK ◽  
Y. PENG ◽  
V. RÖDL ◽  
A. RUCIŃSKI ◽  
...  
Keyword(s):  
Ramsey Number ◽  
Regularity Lemma ◽  
Uniform Hypergraph ◽  
Monochromatic Copy

LetC(3)ndenote the 3-uniformtight cycle, that is, the hypergraph with verticesv1, .–.–.,vnand edgesv1v2v3,v2v3v4, .–.–.,vn−1vnv1,vnv1v2. We prove that the smallest integerN=N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph withNvertices contains a monochromatic copy ofC(3)nis asymptotically equal to 4n/3 ifnis divisible by 3, and 2notherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl.

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