A binary relation $R$ on $\mathbb{N}$ is $(n,k)$-Ramseyan iff for every coloring $f:[\mathbb{N}]^n\rightarrow l$, there exists a subset $G$ of $\mathbb{N}$ such that $(G,R)$ is isomorphic to $(\mathbb{N},R)$ and $|f([G]^n)|\leq k$.

Question: Which binary relation is $(n,k)$-Ramseyan?

Is there any known result?

We know some special example. Say when $R$ is trivial (R=$\emptyset$), then it is $(n,1)$-Ramseyan (which is just the classic Ramsey theorem). When $(\mathbb{N},R)$ is a linear dense order structure then it is $(2,2)$-Ramseyan, $(3,12)$-Ramseyan and for every $n$ there exists $k_n$ such that it is $(n,k_n)$-Ramseyan.

A binary relation $R$ on $\mathbb{N}$ is $(n,k)$-Ramseyan iff for every coloring $f:[\mathbb{N}]^n\rightarrow l$, there exists a subset $G$ of $\mathbb{N}$ such that $(G,R)$ is isomorphic to $(\mathbb{N},R)$ and $|f([G]^n)|\leq k$ (i.e., there exists a 1-1 mapping $h:G\rightarrow \mathbb{N}$ such that $R(x,y)$ if and only if $R(h(x),h(y))$).

Question: Which binary relation is $(n,k)$-Ramseyan?

Is there any known result?

We know some special example. Say when $R$ is trivial ($R=\emptyset$), then it is $(n,1)$-Ramseyan (which is just the classic Ramsey theorem). When $(\mathbb{N},R)$ is a linear dense order structure then it is $(2,2)$-Ramseyan, $(3,12)$-Ramseyan and for every $n$ there exists $k_n$ such that it is $(n,k_n)$-Ramseyan.

A binary relation $R$ on $\mathbb{N}$ is $(n,k)$-Ramsian iff for every coloring $f:[\mathbb{N}]^n\rightarrow l$, there exists a subset $G$ of $\mathbb{N}$ such that $(G,R)$ is isomorphic to $(\mathbb{N},R)$ and $|f([G]^n)|\leq k$.

Question: Which binary relation is $(n,k)$-Ramsian?

Is there any known result?

We know some special example. Say when $R$ is trivial (R=$\emptyset$), then it is $(n,1)$-Ramsian (which is just the classic Ramsey theorem). When $(\mathbb{N},R)$ is a linear dense order structure then it is $(2,2)$-Ramsian, $(3,12)$-Ramsian and for every $n$ there exists $k_n$ such that it is $(n,k_n)$-Ramsian.

A binary relation $R$ on $\mathbb{N}$ is $(n,k)$-Ramseyan iff for every coloring $f:[\mathbb{N}]^n\rightarrow l$, there exists a subset $G$ of $\mathbb{N}$ such that $(G,R)$ is isomorphic to $(\mathbb{N},R)$ and $|f([G]^n)|\leq k$.

Question: Which binary relation is $(n,k)$-Ramseyan?

Is there any known result?

We know some special example. Say when $R$ is trivial (R=$\emptyset$), then it is $(n,1)$-Ramseyan (which is just the classic Ramsey theorem). When $(\mathbb{N},R)$ is a linear dense order structure then it is $(2,2)$-Ramseyan, $(3,12)$-Ramseyan and for every $n$ there exists $k_n$ such that it is $(n,k_n)$-Ramseyan.