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First, we see this example. Suppose we have a set of 6 elements, we can get 3 subsets of it, each of which has 2 elements, but no two sets overlap. But if our set has 5 elements, we want to get 3 subsets of it each of which has 2 elements . Then two of them need to overlap on 1 element. Now generally suppose we have a set of #a elements and we want 3 subsets of it each of which has #a' elements. We also want each two of them have the same but least # in common. Could you get the relationship between a and a'? How many does any two of them in common? How many does three of them has in common? For example, if a=3a', the we can have no pair of the three subsets overlap.

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You might try the problem for dividing a set into two sets first, and explore the possibilities. For working with three sets, I like to visualize the set as evenly distributed around a circle, with the subsets covering certain arcs of the circle. Perhaps this visualization will help you.

Gerhard "Ask Me About System Design" Paseman, 2010.04.11

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What happened to your other middle names? – Will Jagy Apr 12 '10 at 2:30
They are recorded for posterity in the Google archives, the Math Overflow dumps to be, and in other places. Go ahead, I dare you: ask me about System Design. Gerhard "Ask Me About System Design" Paseman, 2010.04.11 – Gerhard Paseman Apr 12 '10 at 2:43
Please tell me about system design – Will Jagy Apr 12 '10 at 2:45
I would be happy to do so. On Google Groups, I have a profile which will reveal an email address to humans. Serious inquiries directed to that address will be honoured and treated respectfully. As a teaser, consider an emergency kit, a list, Euclid's Elements, the recent health care bill passed in Congress, Five Element Acupuncture, solipsism, and the notepad that was tacked on to my mother's refirgerator. What do they all have in common? Gerhard "This Space For Rent" Paseman, 2010.04.11 – Gerhard Paseman Apr 12 '10 at 2:59
I tried one address I found, it was a gmail account and involved three dots as in ... I tried with and without the dots but : This is the Postfix program at host webmail.XXXXXXXXX. I'm sorry to have to inform you that your message could not be delivered to one or more recipients. It's attached below. Does not mean I found the address you really meant, of course. My correct address is readily found through the AMS as in my profile here. My gut response to was – Will Jagy Apr 12 '10 at 3:43

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