Re: Infinite Meet or no-meets


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

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.

But the problem says that we need to find an infinite set such that
either (1) everyone in that set has met everyone else in the set or
(2) no one in the set has met anyone else in the set.

I don't see that your partition helps us do that.

If A is infinite we have satisfied the conditions of the problem.
We have an infinite subset where every member has not met all the
the others in that set.

If A is finite, then its complement B is infinite. B consists of a a
set
where every member does know all the others. Again, the conditions of
the problem are satisfiied.

Perhaps the problem is the wording. "all the people in the subset have
not
met one another."

I take this to mean "each member has not met ALL the others" and not
"noone in the set has met anyone else". The wording is ambiguous.

It is possible for some people in the set to have met some of the
others,
but not all. If there are (say) 5 people and person 1 has met only 2
and 3,
then they have not ALL met one another. You think it means quote:

"(2) no one in the set has met anyone else in the set."

and that is not how I read it.

.

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