# Questions tagged [ramsey-theory]

Branch of combinatorics with the philosophy that 'total disorder is impossible'. For example, Ramsey's theorem asserts that for each $n$, every sufficiently large graph either contains a clique of size $n$ or a stable set of size $n$.

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### Looking for “Set theory for a small universe” by Ketonen

In the paper Partition theorems for systems of finite subsets of integers, Pudlák and Rödl show a Ramsey-type result. The main feature of this result is that the sizes of sets in such systems are not ...
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### The set of homogeneous solutions of a clopen contains an hyperarithmetical set

In the context of Galvin-Prikry generalization of Ramsey's theorem, I read in a couple of papers ([1],[2]) that Solovay [3] proved that if $P$ is a clopen of $[\mathbb{N}]^{\mathbb{N}}$ then the set ...
Problem 1. Is it true that each uncountable group $G$ contains two subsets $A,B\subset G$ such that 1) for any $x,y\in G$ the intersection $xA\cap yB$ is finite and 2) for any function $\... 1answer 224 views ### A combinatorial property of uncountable groups Let$A,B$be two uncountable sets in a group$G$such that for any elements$x,y\in G$the intersection$xA\cap yB$is finite. Let$\Phi:G\to 2^G$be a function assigning to each element$x\in G$some ... 0answers 555 views ###$R(3,6) = 18$, especially proving that$R(3,6)>17$I'm studying the Ramsey numbers, especially$R(3,6) = 18$I understand that the proof using the theorem$R(m,n) <R(m-1,n)+R(m,n-1)$can only prove that$R(3,6)<20$. However by Cariolaro's "On ... 1answer 60 views ### monochromatic induced subgraph in a complete 3-partite graph$H=(V_1\cup V_2 \cup V_3, E)$is a complete$3$partite graph such that$|V_1|=|V_2|=|V_3|=n$. Color the edges with three colors. My question is: Is it possible to find sets$V_1' \subset V_1, V_2' ...
I have found a statement in the introduction of the paper 'Sets of Recurrence and Generalized Polynomials' by Bergelson & Haland, which is Result: Given a polynomial $p \in \mathbb{R}[x]$ such ...