A Ramsey-type problem in directed and bipartite graphs

1973 ◽  
Vol 3 (3-4) ◽  
pp. 299-304 ◽  
Author(s):  
A. Gyárfás ◽  
J. Lehel
Keyword(s):  
Bipartite Graphs ◽  
Type Problem ◽  
Ramsey Type

scholarly journals A Ramsey-type problem on right-angled triangles in space

1996 ◽  
Vol 150 (1-3) ◽  
pp. 61-67 ◽  
Author(s):  
Miklós Bóna ◽  
Géza Tóth
Keyword(s):  
Type Problem ◽  
Ramsey Type

scholarly journals Partitioning the Power Set of $[n]$ into $C_k$-free parts

10.37236/8385 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Eben Blaisdell ◽  
András Gyárfás ◽  
Robert A. Krueger ◽  
Ronen Wdowinski
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Type Problem ◽  
Color Class ◽  
Power Set ◽  
Ramsey Type ◽  
Partition Class ◽  
Cycle Path

We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle — three different subsets $A,B,C\subseteq [n]$ such that $A\cap B,A\cap C,B\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts.  We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp when $G$ is a cycle, path, or star. Additional bounds are given when $G$ is a $4$-cycle and when $G$ is a claw.

scholarly journals On a Ramsey-type problem of Erdős and Pach

2015 ◽  
Vol 49 ◽  
pp. 821-827 ◽  
Author(s):  
Ross J. Kang ◽  
Viresh Patel ◽  
Guus Regts
Keyword(s):  
Type Problem ◽  
Ramsey Type

scholarly journals Finding Unavoidable Colorful Patterns in Multicolored Graphs

10.37236/8184 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Matt Bowen ◽  
Ander Lamaison ◽  
Alp Müyesser
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Type Problem ◽  
Polish Spaces ◽  
Subgraph Isomorphic ◽  
Simple Characterization ◽  
Color Class ◽  
Ramsey Type

We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollobás, concerning colorings of $K_n$ where each color is well-represented. Let $\chi$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $\chi$ $\varepsilon$-balanced if all color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all $\varepsilon$-balanced colorings $\chi$ of $K_n$ contain a subgraph isomorphic to $H$ in its coloring. Bollobás conjectured a simple characterization of $H$ for which $R_\varepsilon^2(H)$ is finite, which was later proved by Cutler and Montágh. Here, we obtain a characterization for arbitrary values of $r$, as well as asymptotically tight bounds. We also discuss generalizations to graphs defined on perfect Polish spaces, where the corresponding notion of balancedness is each color class being non-meagre. 

scholarly journals Note on a Ramsey-type problem in geometry

1994 ◽  
Vol 65 (2) ◽  
pp. 302-306 ◽  
Author(s):  
György Csizmadia ◽  
Géza Tóth
Keyword(s):  
Type Problem ◽  
Ramsey Type

On a Ramsey-type problem

1983 ◽  
Vol 7 (1) ◽  
pp. 79-83 ◽  
Author(s):  
F. R. K. Chung
Keyword(s):  
Type Problem ◽  
Ramsey Type

Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions

2003 ◽  
Vol 12 (5-6) ◽  
pp. 477-494 ◽  
Author(s):  
Noga Alon ◽  
Michael Krivelevich ◽  
Benny Sudakov
Keyword(s):  
Positive Constant ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Ramsey Number ◽  
Absolute Positive Constant ◽  
Turán Number ◽  
Minimum Number ◽  
Ramsey Type ◽  
General Graphs ◽  
Refined Version

For a graph H and an integer n, the Turán number is the maximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is r-degenerate if every one of its subgraphs contains a vertex of degree at most r. We prove that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, . This is tight for all values of r and can also be derived from an earlier result of Füredi. We also show that there is an absolute positive constant c such that, for every fixed bipartite r-degenerate graph H, This is motivated by a conjecture of Erdős that asserts that, for every such H, For two graphs G and H, the Ramsey number is the minimum number n such that, in any colouring of the edges of the complete graph on n vertices by red and blue, there is either a red copy of G or a blue copy of H. Erdős conjectured that there is an absolute constant c such that, for any graph G with m edges, . Here we prove this conjecture for bipartite graphs G, and prove that for general graphs G with m edges, for some absolute positive constant c.These results and some related ones are derived from a simple and yet surprisingly powerful lemma, proved, using probabilistic techniques, at the beginning of the paper. This lemma is a refined version of earlier results proved and applied by various researchers including Rödl, Kostochka, Gowers and Sudakov.

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