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Results tagged with ramsey-theory
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user 39521
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 $n$.
8
votes
Accepted
Ramsey type theorem
Yes, your conjecture is true.
Suppose otherwise. Then there exists a counterexample $f : \mathcal{P}(8) \rightarrow \{0, 1\}$. For each set $X \in \mathcal{P}(8)$, let the proposition $P_X$ denote $f …
- 12.4k
7
votes
1
answer
144
views
Ramsey theory in infinite-dimensional projective spaces
Let $\mathbb{F}_q$ be a finite field and $k$ be a positive integer. If we colour each point of the infinite-dimensional projective space $\mathbb{F}_q \mathbb{P}^{\infty}$ with one of $k$ colours, can …
- 12.4k
4
votes
Accepted
Ramsey theory in infinite-dimensional projective spaces
After further investigation, it appears that disappointingly the answer is 'no', and a proof appears in Lemma 2.4 of Partition Theorems for Subspaces of Vector Spaces (Cates and Hindman, 1975).
- 12.4k
2
votes
Geometric van der waerden theorem
Fedja has already answered one possible interpretation of your question in a comment, where the common ratio is required to be an integer. Here's further explanation, together with a multiplicative an …
- 12.4k