Can a subset be chosen that intersects with other subsets a given number of times?

Let $$S$$ be a set，$$|S| = n$$, with $$n$$ sufficiently large and divisible by $$8$$. Suppose that $$A_1, \dots, A_{n/2} \subseteq S$$, $$|A_i| = \dfrac{n}{2}$$ for all $$i$$. Is it always possible to choose a subset $$T \subseteq S$$ with $$|T| = \dfrac{n}{4}$$ such that $$|T \cap A_i| \geq \dfrac{n}{8}$$ for all $$i$$?

I'm considering a set of integers from $$1$$ to $$n$$. If each $$A_i = \{i, i+1, \dots, i + n/2\}$$, then choosing $$T$$ to be every 4th integer would do the trick; that is, we can let $$T = \{1,5,9,13,\dots\}$$. It would also work to let $$T$$ be spread more coarsely across the integers, for example letting $$T = \{1,2,9,10,17,18,\dots\}$$, or even letting $$T = \{1, \dots, n/8, n/2 + 1, \dots, 5n/8\}$$. I'm trying to see if this idea is true in general.

I'm having problems using a standard counting argument, because as I build $$T$$, I have to keep track of which $$A_i$$'s already intersect with $$T$$ a sufficient number of times and don't need to be considered anymore. Furthermore, I know that the number of sets, $$n/2$$, is critical to the argument, because this would definitely not work if there were, say, $$2^n$$ subsets of $$S$$.

If there are any known results related to this kind of idea, or if anyone has any clever arguments, I would be extremely grateful to hear them. Thank you.

• This would be easier to think about if $X$ looked more like a number and less like the name of a set. Can it be something lower-case from the middle of the alphabet? – timtfj Feb 11 at 0:42
• That's a good point. I changed $X$ to $n$. – Peter Bradshaw Feb 11 at 0:44
• Yeah, I suppose I should write that more clearly. – Peter Bradshaw Feb 11 at 0:55
• Maybe you need to work out what configuration of the $A_i$'s makes it hardest to get the intersections. I think it's to do with how few of them each element of $T$ is in. Roughly speaking,' "how disjoint" the $A_i$'s are. – timtfj Feb 11 at 1:14
• I think you have to find a way to build the $A_i$'s with the minimum reuse of elements, then see whether this can force $T$ to be too big because not enough elements are in enough sets. If that can't happen, and you can prove you've got the worst-case configuration, then T can exist. I've not tried this though and and I won't be surprised if it's got nasty complicating factors. – timtfj Feb 11 at 1:35