Let $W$ be an infinite word over a finite alphabet $\{1,\dots,n\}$ and $k$ a positive integer. An easy application of Van der Waerden's theorem implies the existence of $k$ disjoint and consecutive intervals in $W$ such that the sum of letters in each interval are equal (Edit. See Barber's answer).
On the other hand, Van der Waerden's theorem for the coloring by two colors (which easily implies the general case of every finite coloring) can be derived by this statement without much effort (Edit. See the comments). So I think of it as an equivalent form of original Van der Waerden theorem.
I had conjectured a stronger version which I was unable to prove or disprove it:
Conjecture: If $W$ is an infinite word over finite alphabet. Then for every positive integer $k$, there exists $k$ disjoint and consecutive intervals in $W$, with the same number of occurrence of each letter.
I also know that a weaker version of this conjecture for binary alphabet and "the same" replaced by "proportional" is correct.
Does there exist a similar known result? Can someone give a prove or counterexample?
Best Answer
Your claim is not true.
Two consecutive blocks having the same number of occurrences of each letter (such as the English word "reappear") form what is called an "abelian square". A google search will easily produce a large literature on this subject. The best result (in terms of alphabet size) is due to the Finnish mathematician V. Keranen in 1992: he proved the existence of an infinite word over a 4-letter alphabet having no abelian squares. 4 is best possible, as an easy backtracking argument shows there is no such word over a 3-letter alphabet.
Three consecutive blocks with the same property form an "abelian cube". Dekking proved in 1979 that there is an infinite word over a 3-letter alphabet avoiding abelian cubes. Again, 3 is best possible for alphabet size.
Four consecutive blocks with the same property form an "abelian 4th power". Dekking also proved in 1979 that there is an infinite word over a 2-letter alphabet avoiding abelian fourth powers.
You can easily find references with a google search.