Po-Shen Loh Interview

Diaz-Lopez, Alexander

doi:10.1090/noti1421

Cited by 2 publications

(11 citation statements)

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“…(1,1,1) (1,2,2) (2,1,3) (2,2,4) (3,3,1) (3,4,2) (4,3,3) (4,4,4) The following proposition gives the easy bounds for general n.…”

Section: -Increasing Sequences Of Triplesmentioning

confidence: 99%

“…For example, (3, 3, 9) < 2 (5, 6, 1) < 2 (7, 7, 7) < 2 (7,8,9), but (1, 2, 3) is not 2-less than (1,2,4).…”

Section: -Increasing Sequences Of Triplesmentioning

confidence: 99%

“…The argument starts with the following example, given in its grid representation. This lives in the set [5]× [5]× [4] and contains ten triples. Since 10 = (5×5×4) 1/2 , this is not yet a suitable example.…”

Section: Long 2-comparable Sequencesmentioning

confidence: 99%

“…where the yellow blocks correspond to cuboids with third dimension (0, 4), the blue block corresponds to a cuboid with third dimension (4,5), and the green blocks (a) A length 28 2-comparable sequence of tuples in [9] 3 , given in grid formulation. Observe that 28 = 9 1.516... > 9 3/2 .…”

Section: Note First Thatmentioning

confidence: 99%

“…This paper concerns a deceptively simple problem formulated recently by Po-Shen Loh [7]. As he put it in an interview [4], "I thought it had to be trivial, it's so easy to describe, surely it will fall from some simple argument like the pigeonhole principle, and I will be done. I wasn't done in one hour, actually I'm still not done, and in fact there have been quite a few people who tried it and they also are not done.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Gowers,

Long

2016

Preprint

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We prove a number of results related to a problem of Po-Shen Loh [7], which is equivalent to a problem in Ramsey theory. Let a = (a 1 , a 2 , a 3 ) and b = (b 1 , b 2 , b 3 ) be two triples of integers. Define a to be 2-less than b if a i < b i for at least two values of i, and define a sequence a 1 , . . . , a m of triples to be 2-increasing if a r is 2-less than a s whenever r < s. Loh asks how long a 2increasing sequence can be if all the triples take values in {1, 2, . . . , n}, and gives a log * improvement over the trivial upper bound of n 2 by using the triangle removal lemma. In the other direction, a simple construction gives a lower bound of n 3/2 . We look at this problem and a collection of generalizations, improving some of the known bounds, pointing out connections to other well known problems in extremal combinatorics, and asking a number of further questions.

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“…(1,1,1) (1,2,2) (2,1,3) (2,2,4) (3,3,1) (3,4,2) (4,3,3) (4,4,4) The following proposition gives the easy bounds for general n.…”

Section: -Increasing Sequences Of Triplesmentioning

confidence: 99%

“…For example, (3, 3, 9) < 2 (5, 6, 1) < 2 (7, 7, 7) < 2 (7,8,9), but (1, 2, 3) is not 2-less than (1,2,4).…”

Section: -Increasing Sequences Of Triplesmentioning

confidence: 99%

Section: Long 2-comparable Sequencesmentioning

confidence: 99%

Section: Note First Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Gowers,

Long

2016

Preprint

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show abstract

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

Gowers¹,

Long²

2021

Combinator. Probab. Comp.

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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 a i < b i for at least two values of i, and define a sequence a1, …, a m of triples to be 2-increasing if a r is 2-less than a s 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 trivial upper bound of n2 by using the triangle removal lemma. In the other direction, a simple construction gives a lower bound of n3/2. We look at this problem and a collection of generalizations, improving some of the known bounds, pointing out connections to other well-known problems in extremal combinatorics, and asking a number of further questions.

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Po-Shen Loh Interview

Cited by 2 publications

References 0 publications

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

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

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

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