Consider the following statement (called $\Delta^d(\kappa^+, \lambda)$ where $d\in \omega$ and $\kappa<\lambda$ are cardinals): For every $A': [\lambda]^d\to [\lambda]^{\leq \kappa}$, there exist $E\in [\lambda]^\kappa$ and $A: [\lambda]^{\leq d} \to [\lambda]^{\leq \kappa}$ satisfying
- for any $u\in [E]^d$, $A(u)\supset A'(u)$
- for any $u,v\in [E]^d, type(A(u))=type(A(v))$ and the unique isomorphism between $A(u)$ and $A(v)$ takes $u$ to $v$ and $A'(u)$ to $A'(v)$
- for any $u,v\in [E]^d, A(u)\cap A(v)=A(u\cap v)$
- for $u'\subset u, v'\subset v$ where $u,v\in [E]^d$, if $(u,u',\in)\simeq (v,v',\in)$, then $(A(u), A(u'),\in)\simeq (A(v),A(v'),\in)$.
It's easy to see that when $d=1$ and $\lambda$ regular satisfying $\forall \gamma<\lambda \ \gamma^{\kappa}<\lambda$, this is easily implied by the $\Delta$-system lemma. In fact we get something better, $E\in[\lambda]^\lambda$. When $d>1$, things get a bit trickier. With the right cardinal arithmetic assumption, it can still be made true. In particular, if $\lambda\rightarrow (\kappa)_{2^\kappa}^{2d}$, this is true (due to Shelah in the proof of Lemma 4.1 in Strong partition relations below the power set: consistency, was Sierpinski right, II?).
Note that when $d=1$, we only need $\lambda=(2^\kappa)^+$, which is smaller than the least $\lambda$ such that $\lambda\rightarrow (\kappa)_{2^\kappa}^{2d}$. To me there is some annoying gap between d=1 and d>1 which might suggest improvement for d>1. Is the $\lambda\rightarrow (\kappa)_{2^\kappa}^{2d}$ necessary for $\Delta^d(\kappa^+,\lambda)$? Or maybe such non-uniformity is inevitable (due to the fundamental difference between 1 and numbers > 1). Any suggestions would be appreciated.
