2026-09-18T22:51:29Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/990692026-06-17T14:46:08Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Peter Shor.
Geneson, Jesse (Jesse T.)
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2015-09-29T19:01:24Z
2015-09-29T19:01:24Z
2015
2015
http://hdl.handle.net/1721.1/99069
921857439
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 63-66).
Extremal functions of forbidden sequences and 0 - 1 matrices have applications to many problems in discrete geometry and enumerative combinatorics. We present a new computational method for deriving upper bounds on extremal functions of forbidden sequences. Then we use this method to prove tight bounds on the extremal functions of sequences of the form (12 ... 1)t for 1 >/= 2 and t >/= 1, abc(acb)t for t >/= 0, and avav'a, such that a is a letter, v is a nonempty sequence excluding a with no repeated letters and v' is obtained from v by only moving the first letter of v to another place in v. We also prove the existence of infinitely many forbidden 0 - 1 matrices P with non-linear extremal functions for which every strict submatrix of P has a linear extremal function. Then we show that for every d-dimensional permutation matrix P with k ones, the maximum number of ones in a d-dimensional matrix of sidelength n that avoids P is 20(k) nd-1
by Jesse Geneson.
Ph.D.
66 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.
Bounds on extremal functions of forbidden patterns
Thesis
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<Abstract>Extremal functions of forbidden sequences and 0 - 1 matrices have applications to many problems in discrete geometry and enumerative combinatorics. We present a new computational method for deriving upper bounds on extremal functions of forbidden sequences. Then we use this method to prove tight bounds on the extremal functions of sequences of the form (12 ... 1)t for 1 >/= 2 and t >/= 1, abc(acb)t for t >/= 0, and avav'a, such that a is a letter, v is a nonempty sequence excluding a with no repeated letters and v' is obtained from v by only moving the first letter of v to another place in v. We also prove the existence of infinitely many forbidden 0 - 1 matrices P with non-linear extremal functions for which every strict submatrix of P has a linear extremal function. Then we show that for every d-dimensional permutation matrix P with k ones, the maximum number of ones in a d-dimensional matrix of sidelength n that avoids P is 20(k) nd-1</Abstract>
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