## Bounds on extremal functions of forbidden patterns

##### Author(s)

Geneson, Jesse (Jesse T.)
DownloadFull printable version (4.305Mb)

##### Other Contributors

Massachusetts Institute of Technology. Department of Mathematics.

##### Advisor

Peter Shor.

##### Terms of use

##### Metadata

Show full item record##### 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

##### Description

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. Cataloged from PDF version of thesis. Includes bibliographical references (pages 63-66).

##### Date issued

2015##### Department

Massachusetts Institute of Technology. Department of Mathematics.##### Publisher

Massachusetts Institute of Technology

##### Keywords

Mathematics.