# No $\Delta-$System on a subset of a singular cardinal.

I've been making my way through the new Kunen and I've come across an exercise that I can't work out. The question is this:

Let $\kappa$ be a singular cardinal. Show that there is a collection $A$ of $\kappa$ many two-element subsets of $\kappa$ such that no element of $[A]^\kappa$ forms a $\Delta-$ system. Where $[A]^\kappa$ is the set of subsets of A of size $\kappa$.

Any help would be appreciated (i.e. hints welcome).

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What is $A$ in this context? –  Asaf Karagila Oct 19 '12 at 0:50
@AsafKaragila From context I presume it refers to the (otherwise unnamed) collection of two-element subsets of $\kappa$; I've edited the post slightly to reflect that. cody, if my edits are incorrect, please feel free to revert. –  Steven Stadnicki Oct 19 '12 at 1:10
$\newcommand{\cf}{\operatorname{cf}}$Let $\cf\kappa=\lambda<\kappa$, and let $\langle\alpha_\xi:\xi<\lambda\rangle$ be a strictly increasing sequence cofinal in $\kappa$ such that $\alpha_0=0$. For $\xi<\lambda$ let $K_\xi=[\alpha_\xi,\alpha_{\xi+1})$; then $\kappa=\bigcup_{\xi<\lambda}K_\xi$, and $|K_\xi|<\kappa$ for each $\xi<\lambda$. Let $$A=\bigcup_{\xi<\lambda}[K_\xi]^2\;;$$ clearly $|A|=\kappa$, and I leave to you the straightforward verification that $A$ has no $\Delta$-system of power $\kappa$.