I am dealing with the test of the OBM (Brasilian Math Olympiad), University level, 2017, phase 2.
As I've said at others topic (questions 1, 2, 3, 4 and 6), I hope someone can help me to discuss this test.
The question 6 says:
We'll consider here words on the alphabet $\{a,b\}$: sequences of $a$'s and $b$'s with finite length. We write $u\leq v$ if $u$ is a subword of $v$, it means, we can get $u$ from $v$ by erasing some letters of $v$ (e.g.: $aba\leq abbab$). We say that a word $u$ discerns the words $x$ and $y$ if $u\leq x$ but it's NOT true that $u\leq y$ or vice versa (it's NOT true that $u\leq x$, BUT $u\leq y$).
Let be $m$ and $l$ positive integers. We say that two words $x$ and $y$ are $m$-equivalents if there is NOT $u$ with length $\leq m$ that discern $x$ and $y$.
a) Prove that if $2m\leq l$, so there are two different words $x$ and $y$ with length $l$ that are $m$-equivalents.
b) Prove that if $2m>l$, so two different words $x$ and $y$ with length $l$ cannot be $m$-equivalents.
Well, I've had some ideas and would like to discuss. I'll use the notation $\sim^m$ to indicate $m$-equivalences.
a) Result 1: $\underbrace{baba...baba}_{2m\text{ letters}}\sim^m \underbrace{baba...baba}_{2(m+1)\text{ letters}}$
Proof by induction:
To $m=1$
$ba\sim^m baba$, because the unique subwords of $ba$ and of $baba$ or the same: $b$ and $a$
To $m=2$
$baba\sim^m bababa$, because the unique subwords of length $2$ of $baba$ and of $bababa$ or the same: $bb,ba,ab$ and $aa$.
Consider $\underbrace{baba...baba}_{2m\text{ letters}}\sim^m \underbrace{baba...baba}_{2(m+1)\text{ letters}}$
If a word of length $m+1$ is a subword of $\underbrace{baba...baba}_{2(m+1)\text{ letters}}$, obviously is a subword of $\underbrace{baba...baba}_{2(m+2)\text{ letters}}$.
Conversely, if is subword of $\underbrace{baba...baba}_{2(m+2)\text{ letters}}$,
(i) Is on there $\underbrace{\boxed{baba...ba}ba}_{2(m+2)\text{ letters}}$ or on $\underbrace{ba\boxed{ba...baba}}_{2(m+2)\text{ letters}}$, so is a subword of $\underbrace{baba...baba}_{2(m+1)\text{ letters}}$;
or (ii) There's some letters from the first $ba$ in the beginning and from the last $ba$ in the ending. The others letters (at most $m-1$) are on $\underbrace{ba\boxed{ba...ba}ba}_{2(m+2)\text{ letters}}$, it means, on a word of type $\underbrace{baba...baba}_{2m\text{ letters}}$. By the hypothesis, this part is on a word of type $\underbrace{baba...baba}_{2(m-1)\text{ letters}}$, so the whole subword in on $ba\underbrace{baba...baba}_{2(m-1)\text{ letters}}ba=\underbrace{baba...baba}_{2(m+1)\text{ letters}}$
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Well, let $2m\leq l$ and the words of legh $l$ of type $abababab...$ and $babababa...$. Consider $l$ even and the case odd is similar. So, $abababab...ab$ and $babababa...ba$.
They are $m$-equivalents. In fact, consider a subword $x$ of length $m$ to $ababab...ab$. If it's in the part $\boxed{ababab...a}b$ or in the part $a\boxed{babab...ab}$, so is on $babababa...ba$. If not, it is of type $a\boxed{~~}b$ where the $m-2$ letters of center is on $\underbrace{baba...baba}_{\text{at least } 2(m-1)\text{ letters}}$.
By the result 1, this center is on $\underbrace{baba...baba}_{2(m-2)\text{ letters}}$, so $x\leq \underbrace{ababa...babab}_{2(m-1)\text{ letters}}\leq \underbrace{bababa...ba}_{\text{at least }2m\text{ letters}}$.
The other implication is similar.
b) If $2m>l$, consider a word of length $l$ and note that or there is less than $m$ letters $a$ or less than $m$ letters $b$. Consider less $a$ and take subwords of length $m$ with all the $a$'s of the initial word. We can put $b$'s to "fill" the subwords indicating where the $b$'s are neighbors of the $a$'s. All of the subwords of this type will define all the neighborhoods, so will define whole word. Thus, it's unique, it means, there's no two different words of length $l$ that are $m$-equivalents.
What do you think?
Edit (September, 21)
As you can see at comments following, my initial ideia to b) is not correct. I think this type of subwords will define the blocks of $a$'s or $b$'s and can give us too the amount of $a$'s or $b$'s, but cannot define the word in some cases. I'd like to post here an example:
1) $bbaba$
Subwords of length $3$:
$bba,bbb,baa,bab,aba$
2) $babba$
Subwords of lenght $3$:
$bba,bbb,baa,bab,aba,\boxed{abb}$
It's correct?
I thinked this is interesting, because the subwords of length $\leq 3$ of the first word cannot discern it of the second word, but just a subword of this last.
