2026-09-19T00:00:07Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/990602026-06-16T18:55:33Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Jacob Fox.
Zhao, Yufei
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2015-09-29T19:00:56Z
2015-09-29T19:00:56Z
2015
2015
http://hdl.handle.net/1721.1/99060
921851434
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 171-179).
We extend various fundamental combinatorial theorems and techniques from the dense setting to the sparse setting. First, we consider Szemerédi regularity lemma, a fundamental tool in extremal combinatorics. The regularity method, in its original form, is effective only for dense graphs. It has been a long standing problem to extend the regularity method to sparse graphs. We solve this problem by proving a so-called "counting lemma," thereby allowing us to apply the regularity method to relatively dense subgraphs of sparse pseudorandom graphs. Next, by extending these ideas to hypergraphs, we obtain a simplification and extension of the key technical ingredient in the proof of the celebrated Green-Tao theorem, which states that there are arbitrarily long arithmetic progressions in the primes. The key step, known as a relative Szemerédi theorem, says that any positive proportion subset of a pseudorandom set of integers contains long arithmetic progressions. We give a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Finally, we give a short simple proof of a multidimensional Szemerédi theorem in the primes, which states that any positive proportion subset of Pd (where P denotes the primes) contains constellations of any given shape. This has been conjectured by Tao and recently proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler.
by Yufei Zhao.
Ph.D.
179 pages
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Sparse regularity and relative Szemerédi theorems
Thesis
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<Abstract>We extend various fundamental combinatorial theorems and techniques from the dense setting to the sparse setting. First, we consider Szemerédi regularity lemma, a fundamental tool in extremal combinatorics. The regularity method, in its original form, is effective only for dense graphs. It has been a long standing problem to extend the regularity method to sparse graphs. We solve this problem by proving a so-called "counting lemma," thereby allowing us to apply the regularity method to relatively dense subgraphs of sparse pseudorandom graphs. Next, by extending these ideas to hypergraphs, we obtain a simplification and extension of the key technical ingredient in the proof of the celebrated Green-Tao theorem, which states that there are arbitrarily long arithmetic progressions in the primes. The key step, known as a relative Szemerédi theorem, says that any positive proportion subset of a pseudorandom set of integers contains long arithmetic progressions. We give a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Finally, we give a short simple proof of a multidimensional Szemerédi theorem in the primes, which states that any positive proportion subset of Pd (where P denotes the primes) contains constellations of any given shape. This has been conjectured by Tao and recently proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler.</Abstract>
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