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 A set of subsets (Posted on 2018-01-17)
Create a collection of ten distinct subsets of S = {1, 2, 3, 4, 5, 6} such that:

1. each subset contains three elements,
2. each element of S appears in five subsets, and
3. each pair of elements from S appears in exactly two subsets.

Please explain how you did it.

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 Solution | Comment 1 of 2
{1, 2, 3}, {1, 2, 4}, {1, 3, 5}, {1, 4, 6}, {1, 5, 6}, {2, 3, 6}, {2, 4, 5}, {2, 5, 6}, {3, 4, 5}, {3, 4, 6}

The interesting thing is that every two subsets have at least one number in common. If you connect the subsets that have two numbers in common, then you get the Petersen graph.

123
/|\
/ | \
/  |  \
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/     236     \
/       / \       \
/       |   |       \
/        |   |        \
124-245-+--+-345-135
|      \  |   |  /      |
|       \ |   | /       |
|        \|   |/        |
|         +  +         |
|         |\ /|         |
|         | + |         |
|         |/ \|         |
|      346 256      |
|       /       \       |
|      /         \      |
|     /           \     |
|    /             \    |
|   /               \   |
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146------------------156

Edited on January 17, 2018, 10:36 am
 Posted by Math Man on 2018-01-17 08:19:50

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