1
ADJACENT RAMSEY THEORY
by
Harvey M. Friedman*
Department of Mathematics
Ohio State University
January 23, 2008
July 27, 2010
August 29, 2010
DRAFT
Abstract. We introduce Adjacent Ramsey Theory, which
investigates solutions to the shift equation f(x
1
,...,x
k
) =
f(x
2
,....
,x
k+1
) and the shift inequality f(x
1
,...,x
k
)
≤
f(x
2
,...,x
k+1
), in the context of functions f:N
k
→
N
r
.
Existence theorems are proved, and shown to have strong
metamathematical properties such as unprovability within
Peano Arithmetic. Adjacent Ramsey Theorems represent a new
level of naturalness and simplicity for independence
results at the level of Peano Arithmetic.
1. f(x
1
,...,x
k
) = f(x
2
,...,x
k+1
).
2. f(x
1
,...,x
k
)
≤
f(x
2
,...,x
k+1
).
3. f computable.
4. f limited.
5. f finite.
1. f(x
1
,...,x
k
) = f(x
2
,...,x
k+1
).
TO BE COMPLETED.
We use N for the set of all nonnegative integers, and [t]
for {0,.
..,t1}, t
∈
N.
THEOREM 1.1. For all f:N
k
→
[2], there exist distinct
x
1
,...,x
k+1
such that f(x
1
,...,x
k
) = f(x
2
,...,x
k+1
).
THEOREM 1.2. For all f:N
k
→
[2], there exist x
1
< .
.. < x
k+1
such that f(x
1
,...,x
k
) = f(x
2
,...,x
k+1
).
THEOREM 1.3. For all f:N
k
→
[3], there exist distinct
x
1
,...,x
k+1
such that f(x
1
,...,x
k
) = f(x
2
,...,x
k+1
).
THEOREM 1.4. For all f:N
k
→
[t], there exist x
1
< .
.. < x
k+1
such that f(x
1
,...,x
k
) = f(x
2
,...,x
k+1
).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
We have proved that theorems 1.2, 1.3 correspond to roughly
k fold iterated exponentiation. Theorem 1.1 however has a
small upper bound.
After we obtained our results, we became aware of
Shift graphs and lower bounds on Ramsey numbers rk(l;r), by
D. Duffus, H. Lefmann, and V. Rodl, Discrete Mathematics,
volume 137, Issues 13 (January 1995), 177187
which has considerable overlap with Theorems 1.1  1.4, but
stated in very different language.
THEOREM 1.5. For all even k
≥
1 and f:[k+1]
k
→
[2], there
exist distinct x
1
,...,x
k+1
such that f(x
1
,...,x
k
) =
f(x
2
,...,x
k+1
). For all odd k
≥
1 and f:[k+2]
k
→
[2], there
exist distinct x
1
,...,x
k+1
such that f(x
1
,...,x
k
) =
f(x
2
,...,x
k+1
).
Proof: Fix k
≥
1, k even. Consider the sequence (0,.
..,k1),
(1,.
..,k),(2,.
..,k,0),(3,.
..,k,0,1),.
..,(k,0,.
..,k
1),(0,.
..,k). There are k+2 terms, which is an even number
of terms. Hence the values of f at some adjacent pair must
be equal.
Fix k
≥
1, k odd. Consider the sequence (0,.
..,k1),
(1,.
..,k),(2,.
..,k,k+1),(3,.
..,k,k+1,0),.
..,(k+1,0,.
..,k
2),(0,.
..,k1). There are k+3 terms, which is an even
number of terms. Hence the values of f at some adjacent
pair must be equal. QED
THEOREM 1.6. Theorem 1.5 is best possible in the sense that
k+1 cannot be replaced by k, and k+2 cannot be replaced by
k+1.
Proof: The first claim is obvious. For the second claim,
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 JOSHUA
 Math, Logic, Pythagorean Theorem, Existence, RCA0

Click to edit the document details