A set $\{1,2,3,..,9\}$ is split into two subsets.Prove that there exists one subset,there are $3$ numbers of which one is arithmetic mean of other two

Question

The set $\{1,2,3,..,9\}$ is split into two subsets . Prove that in atleast one subset ,there are $3$ numbers of which one is arithmetic mean of the other two. The solution given in Problem Solving Strategies by Arthur Engel is as follows:

Try to draw a tree with vertices of two colors while avoiding an arithmetic progression. You will not get beyond depth $8$.

However, I am not getting it...I dont know how to draw trees and such stuffs at this level...I mean I know that these things are part of group theory ...if I am not wrong...but of anyone says the basic stuffs for these problem...it would have been very helpful...I know that group theory is a huge part...but the problem applies this....I did not get the idea of the solution

The blockquotes makes the content quite unclear. Is the inner blockquote your statement, or is it cited from the material you cited? — soupless, Mar 28 '22 at 14:18
Hint: Any partition of ${1,2,\ldots,9}$ must contain a $3$-Term Arithmetic Progression is highly relevant. — MJD, Mar 28 '22 at 14:18
Does this answer your question? Any partition of ${1,2,\ldots,9}$ must contain a $3$-Term Arithmetic Progression — JMP, Mar 28 '22 at 14:27
@JMP that post shows how to do the search effectively, but as I described below, actually doing the search is something you can only do when you already understand what is being searched and why. It appears from the question that OP doesn't understand that yet, so it seems to me that this is not really a duplicate of the other one. — MJD, Mar 28 '22 at 14:32
@JMP no , i have checked those answers but they are not quite detailed and according to me kind of a vague answer ...and since I cant comment everywhere as of now according to the rules of the site....so I reposted the question specifically asking for the portion I need a clarification upon.. — , Mar 28 '22 at 14:38

MJD · Answer 1 · 2022-03-28T14:36:05.967

The suggestion in the book is very unclear. I believe the idea it is suggesting it this:

Say the two sets are called R and B. One of these sets contains “3 numbers of which one is arithmetic mean of the other two” precisely when it contains 3 numbers $a, a+d, a+2d$ in arithmetic progression, since $a+d$ is the mean of $a$ and $a+2d$.
Imagine the the elements of set $R$ are colored red and the elements of set $B$ are colored blue. Splitting $\{1, \ldots 9\}$ into two subsets can now be understood as coloring the elements red and blue.
There are not that many ways to assign the colors: $2^9=512$ ways, or $256$ if you take advantage of the obvious symmetry.
So if there is a counterexample to the claim — a coloring of the nodes where there is no arithmetic progression of a single color — it will not take long to find, even with a brute-force search.
Searching can naturally be understood as traversing a tree. In this example the root node of the tree has all the elements of $\{1, \ldots 9\}$ uncolored. When descending to a child node, color one of the elements either red, if going to the left, or blue, if going to the right. In a leaf node, 9 steps down, all nine elements have been colored.
Delete every node from the tree that does have an arithmetic progression of a single color.
If any leaf node remains, that is your counterexample.
But it is not hard to show that every leaf node will be deleted; therefore there is no counterexample.

If you are studying problem-solving, each of these ideas is worth considering carefully. Some of them (think of partitions as colorings; think of search as tree traversal) are very general and come up in many contexts.

A set $\{1,2,3,..,9\}$ is split into two subsets.Prove that there exists one subset,there are $3$ numbers of which one is arithmetic mean of other two

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