Manymathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemeredi’s theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep bymanymathematicians, and somakes for a good case on which to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered. Mathematicians frequently cite depth as an important value for their research. A perusal of the archives of just the Annals of Mathematics since the 1920s reveals more than a hundred articles employing the modifier ‘deep’, referring to deep results, theorems, conjectures, questions, consequences, methods, insights, connections, and analyses. However, there is no single widely shared understanding of what mathematical depth consists in. This article is a modest attempt to bring some coherence to mathematicians’ understandings of depth, as a preliminary to determining more precisely what its value is. I make no attempt at completeness here; there are many more understandings of depth in the mathematical literature than I have categorized here, indeed so many as to cast doubt on the notion that depth is a single phenomenon at all. Yet I hope to advance our philosophical understanding of this rich cluster of values a little bit, perhaps making way for more unified accounts to follow. My strategy in this article is to introduce Szemeredi’s theorem, that every sufficiently ‘dense’ subset ofN contains an arbitrarily long arithmetic progression, as a case study for focusing an investigation of mathematical depth. After discussing the content †Thanks to Jeremy Heis, Penelope Maddy, Bennett McNulty, Sean Walsh, and Jim Weatherall for organizing the conference on mathematical depth at UC Irvine in April 2014 at which this paper was first presented, and to the participants of this conference for stimulating discussions. Thanks to Jordan Ellenberg, Balazs Gyenis, Juliette Kennedy, and Jonathan Livengood for helpful comments on drafts of the article. Philosophia Mathematica (III) Vol. 00 No. 0 C ©The Authors [2015]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com • 1 Philosophia Mathematica Advance Access published January 6, 2015