On a plane, a set of circles is called a
chain
if for any two circles in the set, one is completely inside the other. Note that any subset of a chain is also a chain.
There are 100 (distinct) circles on the plane. Among any 10 of them, at least two circles form a chain. What is the minimum number of chains of size 10?
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Let a chain of 1 0 circles be called a bundle .
Note that if we have a chain of n circles, then the number of chains of size k in this chain is simply ( k n ) ; pick any k circles and they form a chain.
First, we claim that this number is at most 1 5 4 . Consider a collection of circles: 1 2 circles form a single chain, and the remaining 8 8 circles are divided into 8 chains of 1 1 circles each; these 9 chains are otherwise disjoint. By pigeonhole principle , there are 1 0 circles picked but only 9 chains present, so there are two circles that come from the same chain, and thus they form a chain, satisfying the condition of the problem. Afterwards, we can easily see that there are ( 1 0 1 2 ) = 6 6 bundles in the chain of size 1 2 , and ( 1 0 1 1 ) = 1 1 bundles in any chain of size 1 1 , for a total of 6 6 + 8 ⋅ 1 1 = 1 5 4 bundles. Finally, there is no other bundle; picking two circles from different chains won't make them a chain.
We will now prove that 1 5 4 is indeed the minimum (any such collection of circles must have at least 1 5 4 bundles).
Consider a partial order ( S , ≤ ) where S is the set of circles on the plane and ≤ is the relation completely contained in ; that is, if a ≤ b , then a is completely contained in b . It's straightforward to see that ≤ is a partial order:
Recall that if a ≤ b or b ≤ a , then a and b are said to be comparable ; otherwise, they are incomparable . A chain is thus an actual chain in order theory terminology: a set of elements such that any two of them are comparable. Also recall that an antichain (or independent set ) is a set of elements such that any two of them are incomparable.
The condition of the statement states that there is no antichain of size 1 0 , for if there is any, those 1 0 circles are pairwise incomparable, contradicting that some two circles should form a chain and thus comparable. Thus the size of the largest antichain is 9 . By Dilworth's theorem , this implies that it's possible to partition the circles into at most 9 (disjoint) chains; call these chains as big chains .
Suppose there is a big chain of 1 3 or more circles. Then there are at least ( 1 0 1 3 ) = 2 6 4 > 1 5 4 bundles, already passing our target.
Now suppose that there are a big chains of size 1 2 , b big chains of size 1 1 , and c big chains of size 1 0 or less. Also suppose there are B bundles in total. We have:
Note that by pigeonhole principle, there are 1 0 0 circles and 9 big chains, so at least one big chain has 1 2 circles. This means a ≥ 1 . Also,
1 0 0 − 1 2 a = 1 1 b + 1 0 c = b + 1 0 ( b + c ) ≤ b + 1 0 ( 9 − a )
Simplifying, this gives b ≥ 1 0 − 2 a . Also, we know that b ≥ 0 , so we have b ≥ max { 0 , 1 0 − 2 a } . Substituting to the last inequality, we have:
B ≥ ( 1 0 1 2 ) a + ( 1 0 1 1 ) b = 6 6 a + 1 1 b ≥ 6 6 a + 1 1 ⋅ max { 0 , 1 0 − 2 a } ≥ 6 6 a + 1 1 ⋅ ( 1 0 − 2 a ) = 1 1 0 + 4 4 a ≥ 1 1 0 + 4 4 ⋅ 1 = 1 5 4
This proves the claim, showing that 1 5 4 bundles is the minimum.