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