Finite Analogs of Szemerédi's Theorem
By Paul Raff and Doron Zeilberger
[Appeared in "Gems in Experimental Mathematics"
(Contemporary Mathematics series (AMS) v. 517 (2010), T. Amdeberhan, L. Medina, and V. Moll, eds., 313319.]
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Written: July 13, 2009.
One of the "deepest" theorems in mathematics is Endre Szemerédi's
theorem about the inevitability of arithmetical progressions.
Here we try to nibble at it, by doing "finite" analogs.
This is already interesting for its own sake, but
we believe that it has the potential to lead to
extermely interesting sharpening of the currently
rather weak bounds. Let's hope!
Maple and Mathematica Packages
Important: This article is accompanied by
Sample Input and Output for the Maple package ENDRE

In order to get the statements for the explicit expressions for R_{3,d}(N),
the size of the maximal subset of [1,N] avoiding 3term arithmetical progressions
with spacings ≤ d+1, for d between 0 and 6
the input
gives the output.

In order to get the statements for the explicit expressions for R_{4,d}(N),
the size of the maximal subset of [1,N] avoiding 4term arithmetical progressions
with spacings ≤ d+1, for d between 0 and 4
the input
gives the output.

In order to get the generating functions (in t), as well as the first 30 terms,
and the estimated asymptotics, of the counting sequences enumerating
nletter binary words w[1]...w[n] such that there does not exist an i between 1 and n and
a d between 1 and D, such that w[i]w[i+d]w[i+2d]w[i+3d]=1010
(D=1..5)
the input
gives the output.

In order to get the generating functions (in t), as well as the first 30 terms,
and the estimated asymptotics, of the counting sequences enumerating
nletter binary words w[1]...w[n] such that there does not exist an i between 1 and n and
a d between 1 and D, such that neither
w[i]w[i+d]w[i+2d]w[i+3d]=0000 nor
w[i]w[i+d]w[i+2d]w[i+3d]=1111 ,
(D=1..4)
the input
gives the output.

In order to find out which of the generalized single patterns of length 3
(with spacings between 0 and 5) is hardest to avoid, and which is easiest to avoid,
as well as complete generating functions, and asymptotics, for each such pattern
the input
gives the output.

In order to find out which of the generalized single patterns of length 4
(with spacings between 0 and 3) is hardest to avoid, and which is easiest to avoid,
as well as complete generating functions, and asymptotics, for each such pattern
the input
gives the output.

In order to find out which of the generalized single patterns of length 5
(with spacings between 0 and 2) is hardest to avoid, and which is easiest to avoid,
as well as complete generating functions, and asymptotics, for each such pattern
the input
gives the output.

In order to get a completely computergenerated but HUMANREADABLE
statement and PROOF of the explicit expression (as a piecewise linear function)
in n, for the number of nletter binary words, in the alphabet {0,1} that
avoid 111, 1_1_1, and 1_ _ 1 _ _ 1, the
the input
gives the output.
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