Ten subsets of [0, 1]

[bonanova]

I was playing with the following discrete puzzle and wondering if it has a continuous extension:
n balls numbered 1-n are placed in each of 10 urns.
A certain number m of balls is removed from each urn.
The remaining balls have the following property:

1. if any 5 urns are emptied into a bowl, there will be at least one ball with each of the numbers 1-n
2. if any 4 urns are emptied into a bowl, there will be at least one missing number.

The problem asks what is the smallest n that makes this possible, and what is the value of m?
It does not ask which m balls are removed from the urns - a different subset is obviously removed from each urn.

I reasoned that if 10 unique subsets of the set [1, ..., n] can be found with these two conditions for a finite n,
there should be 10 unique subsets of the reals on [0, 1] with the same properties:
the union of any 5 subsets covers the interval, but of any 4 does not.

Maybe it's trivial, but I couldn't find a starting point. Any thoughts?

OK all i had to do is post this and I see the answer.
So let's add the question: what parts of the interval are removed in the ten cases?
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[ralphmerridew]

Minimal n is 10C6 = 210.

If any ball appears in 5 or fewer urns, then choose the other 5 urns and you'll miss that ball. If any ball appears in 7 or more urns, remove that ball and the result will have fewer balls but still have properties 1 and 2. Therefore every ball appears in exactly 6 urns.

Consider any set of six urns. There must be at least one ball that appears in exactly those 6 urns, or choosing the other 4 would get all the balls. There must be at most one, or one of them could be removed without disturbing properties 1 or 2.

Therefore, n = 10C6 in a minimal solution.

[Zag]

I was going to say 10C4, for exactly the same reason (which is, of course, the same answer). My thinking was that for every group of 4 urns, there is exactly 1 ball missing, which must be in all the remaining urns.