2 tag
Post Made Community Wiki by ilyaraz
1

# Density Ramsey theorems with explicit asymptotics

I wonder what interesting and non-trivial examples of density Ramsey theorems with explicit asymptotics are there?

I'm aware of two examples: Szemerédi's theorem and density Hales-Jewett theorem.

Let me try to formalize what I mean by "density Ramsey theorems". I'm interested in statements of the following type: for every $k \in \mathbb{N}, \alpha > 0$ there exists $n_0 = n_0(k, \alpha)$ such that if we have a collection of $n \geq n_0$ objects, then whatever $\alpha n$ objects we take there are $k$ among them that form a "nice" structure. Collections and niceness are problem-specific of course: in case of Szemerédi's theorem they are $\lbrace 1, 2, \ldots, n\rbrace$ and arithmetic progressions, for Hales-Jewett — $[k]^d$ and combinatorial lines, respectively. I'm particularly interested in theorems where the dependence $n_0 = n_0(k, \alpha)$ can be made explicit.