The length of an s-increasing sequence of r-tuples

@article{Gowers2021TheLO,
  title={The length of an s-increasing sequence of r-tuples},
  author={William T. Gowers and Jason Long},
  journal={Comb. Probab. Comput.},
  year={2021},
  volume={30},
  pages={686-721}
}
We prove a number of results related to a problem of Po-Shen Loh [9], which is equivalent to a problem in Ramsey theory. Let a = (a1, a2, a3) and b = (b1, b2, b3) be two triples of integers. Define a to be 2-less than b if ai < bi for at least two values of i, and define a sequence a1, …, am of triples to be 2-increasing if ar is 2-less than as whenever r < s. Loh asks how long a 2-increasing sequence can be if all the triples take values in {1, 2, …, n}, and gives a log* improvement over the… 

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References

SHOWING 1-10 OF 29 REFERENCES
On Sets of Integers Which Contain No Three Terms in Arithmetical Progression.
  • R. Salem, D. Spencer
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1942
example there does not exist a sequence of domains Dl, D2, D3, . .. closing down on the point 0 and such that, for each n, the boundary of D. is compact. ,'Jones, F. B., "Concerning Certain
On Sets of Integers Which Contain No Three Terms in Arithmetical Progression.
  • F. Behrend
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1946
TLDR
By a modification of Salem and Spencer' method, the better estimate 1-_2/2log2 + e v(N) > N VloggN is shown.
An Extension Of The Ruzsa-Szemerédi Theorem
TLDR
An extension of the Erdős, Frankl, Rödl Theorem (and thus the Ruzsa–Szemerédi Theorem) is given to show that indeed the Brown, Erdœs, T. Sós Theorem is not far from being best possible.
1-color-avoiding paths, special tournaments, and incidence geometry
We discuss two approaches to a recent question of Loh: must a 3-colored transitive tournament on $N$ vertices have a 1-color-\emph{avoiding} path of vertex-length at least $N^{2/3}$? This question
Large Subgraphs in Rainbow‐Triangle Free Colorings
TLDR
A generalisation of the celebrated theorem of Erd\H{o}s-Szekeres, which states that any sequence of $n$ numbers contains a monotone subsequence of length at least $\sqrt{n}$.
Packings of Rn by certain error spheres
  • S. Stein
  • Mathematics
    IEEE Trans. Inf. Theory
  • 1984
TLDR
It is shown that packings by the cross that are extremely regular (lattice packings) do just about as well as arbitrary packings in all dimensions and for k large, and for the semicross, even in R^{3} , when the arm length k is large, latticePackings are much less dense than arbitraryPackings.
Directed paths: from Ramsey to Ruzsa and Szemerédi
Starting from an innocent Ramsey-theoretic question regarding directed paths in tournaments, we discover a series of rich and surprising connections that lead into the theory around a fundamental
...
...