Re: Infinite Meet or no-meets


Jesse F. Hughes wrote:
"Pubkeybreaker" <Robert_silverman@xxxxxxxxxxxx> writes:

Guy L. wrote:
I am a bit perplexed by the following problem.

Suppose a set of aleph_0 many people are given. Show there is an
infinite subset of that set such that either all the people in the
subset have met one another or all the people in the subset have not
met one another.

How does one go about to prove the above problem? I would ask my
teachers, but none of my teachers at my high school would know...

Let S be your set. Partition it into two disjoint subsets A and B. So
S = A U B.
Let A be the set of people who have not met one another.

I don't see that your set A is well defined. Suppose that our set
consists only of three people (rather than aleph_0). Suppose persons
1 and 2 have met, but no other pair of persons have met.

Then 1 hasn't met 3, so I guess both 1 and 3 are in the set. But 2
hasn't met 3 either, so is 2 in the set or not?


A is the set of all people such that each person in the set has not met
all
others in the set. So 2 is in the set. In your example, 1,2, and 3
are in the
set since none of them have met all the others. It does not matter if
some
people in the set have met some of the others.

.

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