NotAPerson

Wouldn't the subset {(a, b) | a,b > 20} be monochromatic?

Ed:

Perhaps I phrased it poorly, but I think the point was to show that you can always construct a predicate, P over a and b, such that P(a, b) is finitely defined (such as "a > 20 and b > 20"), but {(a, b) | P(a, b) is true} is infinite and monochromatic.

Instead of having some cases of colorings where your only option is to construct things of the form "(a = 5 and b = 17) or (a = 3 and b = 47) or ..." where you just list out every pair that matches (in an infinite subset).

> More surprisingly perhaps, if you colour the infinite subsets of the natural numbers red or blue, then there exist colourings for which there is no monochromatic subset.

Could you elaborate on this?

The hard part is to show that for any coloring, there's some infinite subset without relying on "well, we can just pick one for each of these infinitely many numbers".

I'm pretty sure that the challenge is to prove that you can for any coloring construct a finitely defined rule for picking the members of the subset which is guaranteed to give you a monochromatic subset.

The question is more about if you can always find such a (finite) rule to partition the set, rather than if you can in a few easily constructed examples.

Imagine you have one person for each (positive) integer, given a unique integer ID at birth, and a hotel with countably infinite rooms, each with a unique room number.

The hotel could have someone in everyone room if every person with an even number as their ID was staying there. That is, for every room number, twice the room number is a unique even number, so there's a 1-to-1 correspondence between the number of rooms and the even integers.

You could then have someone with an odd number ID show up looking for a room, and would have to rearrange from a stay-in-half-your-ID lineup.

It's a (arguably defining) property of infinite sets that they contain a strict subset (at least one guy not in the subset) with the same "size" as the whole set.

So the evens and the integers are an example of this, with both having the same "size", even though the evens are contained in the integers.

> Job seekers are much better served spending an hour sending 10 focused applications than spending that same hour to send 100 sanitized applications.

Let's assume that you have a 1 in 20 chance on the focused applications, and a 1 in 200 chance on the sanitized ones.

Then the odds that you are offered at least one job, is 40.13% for the focused ones, and 39.42% for the sanitized ones.

I'm not sure the emotional investment is enough to justify the difference in likelihood of landing a job.

> Their age doesn't predict their accuracy.

I was actually asking about the case where the demographic they're predicting the opinion of is age restricted -- are they good at predicting the views of a particular age range, even if they're not good at predicting the overall opinion?

My question was just about restricting the age to be near theirs.

This might be an off topic question, but how do the results change (if at all) when accounting for the age of the people involved?

That is, how do politicians do at predicting the opinions of people +/- 5 years of their own age (which may be different than the total for the district)?

I've always been curious if politicians are just "behind the times" (due to an average age higher than the average age of constituents), leading to systemically conservative views.