Question

A theorem that inspired this field implies that, for a party with 6 people, there must be a group of 3 mutual acquaintances or a group of 3 mutual strangers. For 10 points each:
[10m] Name this field, whose key results include the Hales-Jewett theorem. This field is sometimes thought of as an extension of combinatorics.
ANSWER: Ramsey theory
[10e] More abstractly, Ramsey's theorem asserts the existence of a minimum number of vertices that guarantees the existence of monochromatic cliques in a “complete” one of these structures.
ANSWER: graphs
[10h] This major result in Ramsey theory states that, given two integers k and p, there is some number W such that coloring the first W natural numbers with k colors yields a color with an arithmetic progression with p terms. This theorem's namesake developed a way of using dotted or undotted indices to denote right or left-handed spinors.
ANSWER: van der Waerden's theorem [or van der Waerden notation or Bartel Leendert van der Waerden]
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Summary

2023 BHSU @ Berkeley03/18/2023Y316.67100%33%33%
2023 BHSU @ Maryland03/11/2023Y316.67100%33%33%
2023 BHSU @ Northwestern02/25/2023Y620.00100%100%0%
2023 BHSU Online04/15/2023Y417.50100%75%0%
2023 BHSU @ Sheffield04/15/2023Y215.00100%50%0%
2023 BHSU @ Waterloo04/15/2023Y316.67100%67%0%
2023 BHSU @ Yale04/08/2023Y316.67100%67%0%

Data

Maryland AFan-tastic Negs and Where to Find Them10101030
Maryland B+Rutgers Diaspora010010
The Perfection of Wisdom in 8,000 NegsGlass Onyen: A Knives Out Mystery010010
Amartya Senpai Notice MeIllinois1010020
Brownstein et al.Mojo Shojo1010020
Chicago AOSU1010020
WUSTLMichigan1010020
Epic GamesNorthwestern1010020
Chicago B+Purdue University Fightin' Hadrien Davids1010020
Cornell B-Cornell C++1010020
Dr. Christopher Manners, MD – Samarra, IraqRollin in from Sturgis, The Cheyne Gang1010020
Houston and friendsWhat is BH! And you are asking me? BHSU010010
Hutto HipposThe Enrique Trastamara Military Academy B1010020
Berkeley A[Insert Lawyer Joke Here]0101020
Stanford AIt's Joever1010020
Berkeley BStanford B010010
Magisters LudiBroken Imperial Nomads010010
Buzzers KaramazovOxford Hornets1010020
Parth et al.Cameron et al.010010
Neg 5! Playing Quizbowl, Liddy StyleMilan et al.1010020
The Passion According to C.H.The Vulture's Guide to Buzz1010020
UG Championship Players (and Mazin)Columbia B1010020
Watching Arthur Delot-Vilain at Chicago Open Made Me Like French People Againmeet the new weird, same as the old weird1010020
nats is supposed to mean you all have gone softOlmo (Bonus) Bagelry010010