How to prove some properties of partitions of finite subsets of N?

Asked Jan 04 '13 at 16:58

Active Oct 30 '17 at 06:23

Viewed 48 times

Possible Duplicate:
Any partition of {1,2,..,9} must contain a 3-Term Arithmetic Progression

The problem is as such:

Prove that there is not a partition of $N_9 = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ consisting of two sets (the set $N_9$ could not be partitioned into two sets), such that both sets do not contain three elements of an arithmetic progression ($a_n = a_0 + (n-1)d$). Such a partition exists for $N_8 = \{1,2,3,4,5,6,7,8\}$.

A basic example of a partition of $N_9$ is $P = \{\{1,3,4\}, \{2,5,6,7,8,9\}\}$. It is evident that in this case, there are three elements in the second set that belong to the same arithmetic progression. Leaving examples aside, I have no idea on how a proof for this property would look like. Could anyone assist in generating a proof for this property?

edited Oct 30 '17 at 06:23

Peter Taylor

13,425

asked Jan 04 '13 at 16:58

You found a counterexample – Amr Jan 04 '13 at 17:01
@Amr: no, that's just an example – SBF Jan 04 '13 at 17:02
@Amr That is not a counterexample. In the second set, 2, 5, 8 are elements of an arithmetic progression, and also 5, 6, 7, 8, 9 are elements of another arithmetic progression. – Jan 04 '13 at 17:03
@Amr: The second set contains the progression ${5,7,9}$. – Brian M. Scott Jan 04 '13 at 17:03
@Ilya Yes Now I see – Amr Jan 04 '13 at 17:03
@BrianM.Scott: as well as ${5,6,7}$ :) – SBF Jan 04 '13 at 17:04
Alex, notice that if $d = 3$ then the 3rd member shall be at least $10$, so you shall only take care of $d = 1,2$. The example for $N_8$ is ${1,3,4,6}\cup{2,5,7,8}$ – SBF Jan 04 '13 at 17:05
@Ilya I don't see what is your point, could you explain a bit more? – Jan 04 '13 at 17:07
@Alex: nevermind, that part of the comment is incorrect – SBF Jan 04 '13 at 17:10
Feeling stupid here, but definition should be $a_n = a_{n-1} + d$, not the above, right? – gnometorule Jan 04 '13 at 17:11
@gnometorule Yes, I made a mistake. I corrected it to provide a non-recursive definition for the progression. – Jan 04 '13 at 17:12

How to prove some properties of partitions of finite subsets of N?

0 Answers0