# 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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### Happy ending problem - why not a proof by induction? (cont)

After sharing ideas on this post, I have been thinking for some time on the problem, and I think that a possible way to prove the Erdös-Szekeres conjecture could be structured as follows: Consider ...
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### Higher-dimensional Sierpiński partitions

Given a well-ordering of $\mathbb{R}$, there is a natural way to define an associated partition of pairs of real numbers into two pieces: one assigns the value $0$ to a pair $r<s$ if the well-...
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### Dense triangle-free graphs and their independent sets

Recall that a graph is triangle-free if it does not contain a copy of $K_3$. Also, for a graph $G$, $\alpha(G)$ shall denote its independence number. Lastly, we will write $o(1)$ to denote quantities ...
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### Does $2^{\aleph_0}\rightarrow [\aleph_1]^2_3$ require that the continuum is weakly inaccessible?

A classic result of Sierpiński shows that $2^{\aleph_0}\nrightarrow [\aleph_1]^2_2$, that is, there is a coloring of pairs of real numbers using two colors such that both colors appear on any ...
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### Original Paper for "Bipartite Ramsey Theory" by Hattingh, Johannes H, 1998

I'm trying to find the original paper "Bipartite Ramsey Theory" by Hattingh, Johannes H., Util. Math. 53 (1998), 217–230. However, I couldn't find it online except Mathsci. Does anyone ...
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### Happy ending problem – Why not a proof by induction?

I have been thinking for a while on the happy ending problem, looking for approaches to attack the Erdős–Szekeres conjecture: the smallest number of points for which any general position arrangement ...
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### Roelcke precompactness and Ramsey property

A survey by Nguyen Van Thé (2014) has Conjecture 1, which is that "every closed oligomorphic subgroup of $S_∞$ should have a metrizable universal minimal flow with a generic orbit." Later, ...
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### Ramsey style theorem with unbounded colors

Question: Let $\varepsilon>0$ and $N\in\omega$ be sufficiently large (depending on $\varepsilon$). Let $h:\subseteq N\rightarrow N$ be such that $h(B)\notin B$ for all $B\subsetneq N$. Must there ...
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### 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 ...
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