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

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.

Download Full-text

Independence numbers of locally sparse graphs and a Ramsey type problem

Random Structures and Algorithms ◽

10.1002/(sici)1098-2418(199610)9:3<271::aid-rsa1>3.0.co;2-u ◽

1996 ◽

Vol 9 (3) ◽

pp. 271-278 ◽

Cited By ~ 17

Author(s):

Noga Alon

Keyword(s):

Type Problem ◽

Sparse Graphs ◽

Ramsey Type ◽

Independence Numbers

Download Full-text

On a Ramsey-type problem of Erdős and Pach

Electronic Notes in Discrete Mathematics ◽

10.1016/j.endm.2015.06.049 ◽

2015 ◽

Vol 49 ◽

pp. 821-827 ◽

Cited By ~ 1

Author(s):

Ross J. Kang ◽

Viresh Patel ◽

Guus Regts

Keyword(s):

Type Problem ◽

Ramsey Type

Download Full-text

On a ramsey-type problem of J. A. Bondy and P. Erdös. I

Journal of Combinatorial Theory Series B ◽

10.1016/0095-8956(73)90035-x ◽

1973 ◽

Vol 15 (1) ◽

pp. 94-104 ◽

Cited By ~ 68

Author(s):

Vera Rosta

Keyword(s):

Type Problem ◽

Ramsey Type

Download Full-text

Finding Unavoidable Colorful Patterns in Multicolored Graphs

The Electronic Journal of Combinatorics ◽

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.

Download Full-text