# On a combinatorial set covering property

Let $$\kappa < \lambda < \mu$$ be infinite cardinals. Is there a collection $${\cal U}\subseteq {\cal P}(\mu)$$ of subsets of $$\mu$$ with the following properties?

1. for all $$U\in {\cal U}$$ we have $$|U| = \lambda$$;
2. every $$S\subseteq \mu$$ with $$|S| = \kappa$$ is contained in exactly one member of $${\cal U}$$; and
3. for all $$\alpha,\beta \in \mu$$ we have $$|{\cal U}_\alpha| = |{\cal U}_\beta|$$, where $${\cal U}_\alpha = \{U\in {\cal U}: \alpha\in U\}$$, and $${\cal U}_\beta$$ is defined similarly.
• If $\kappa$ is infinite, then no. Let $V$ be a $\kappa$ sized subset of $U$, and let $W$ have one more element than $V$ with that element outside of $U$. If there is $U'$ containing $W$, then both $U$ and $U'$ contain $V$, violating condition $2$. Gerhard "Not Quite A Projective Plane" Paseman, 2018.12.18. – Gerhard Paseman Dec 18 '18 at 15:49
• Isn't 3) implied by 2)? Just enumerate all $\kappa$-sets containing $\alpha$... – Ilya Bogdanov Dec 18 '18 at 18:32

Extending the problem, admit the possibility of equality of cardinals, so consider that $$\kappa \leq \mu$$. If indeed $$\kappa=\mu$$, then the collection of subsets with one member, namely $$\mu$$, suffices. But that is the only case.
For we borrow the argument from the comments. Let $$U$$ be a proper subset of $$\mu$$ that is part of a suitable collection, and let $$V$$ be $$\kappa$$-sized and contained in $$U$$. If $$\kappa$$ is infinite, pick some element of $$\mu \setminus U$$ and consider a set $$U'$$ from the collection that contains the $$\kappa$$-sized set which is $$V$$ union (the singleton set containing) this element. Then both $$U$$ and $$U'$$ contain $$V$$, which means condition two does not hold, and so a suitable collection cannot have a proper subset of $$\mu$$.
For finite $$\kappa$$ and infinite $$\lambda \lt \mu$$, here is an idea which may provide a suitable collection, but some work needs to be done. Namely, well order all of the $$\lambda$$-sized subsets of $$\mu$$, and pick the least allowed $$\lambda$$ sized subset $$U$$, and then "throw out" the $$\lambda$$ sets that intersect $$U$$ in a subset of size at least $$\kappa$$. Conditions 1) and 2) are partly satisfied during the construction, but it remains to show that every $$\kappa$$ subset is uniquely covered. Since $$\mu$$ is strictly larger than $$\lambda$$, this should be possible to show.
• If we consider $\kappa=1$, the picture changes, and any equipartition of $\mu$ works. Things get interesting for larger $\kappa$, and it is unclear to me what a suitable collection would be even for $\kappa=2$. Probably this problem was considered by Erdos, but I have no references for you. (And, you are welcome!) Gerhard "Finitely Unclear, My Dear Dominic" Paseman, 2018.12.20. – Gerhard Paseman Dec 20 '18 at 16:48