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TZOFFSETFROM:+0900
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DTSTART:20240101T000000
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DTSTART;TZID=Asia/Seoul:20250805T163000
DTEND;TZID=Asia/Seoul:20250805T173000
DTSTAMP:20260416T160801
CREATED:20250713T060700Z
LAST-MODIFIED:20250725T225751Z
UID:11148-1754411400-1754415000@dimag.ibs.re.kr
SUMMARY:Tony Huynh\, Rainbow triangles and the Erdős-Hajnal problem in projective geometries
DESCRIPTION:We formulate a geometric version of the Erdős-Hajnal conjecture that applies to finite projective geometries rather than graphs.  In fact\, we give a natural extension of the ‘multicoloured’ version of the Erdős-Hajnal conjecture. Roughly\, our conjecture states that every colouring of the points of a finite projective geometry of dimension $n$ not containing a fixed colouring of a fixed projective geometry $H$ must contain a subspace of dimension polynomial in $n$ avoiding some colour. \nWhen $H$ is a ‘triangle’\, there are three different colourings\, all of which we resolve.  We handle the case that $H$ is a ‘rainbow’ triangle by proving that rainbow-triangle-free colourings of projective geometries are exactly those that admit a certain decomposition into two-coloured pieces. This is closely analogous to a theorem of Gallai on rainbow-triangle-free coloured complete graphs. The two non-rainbow colourings of $H$ are handled via a recent breakthrough result in additive combinatorics due to Kelley and Meka.  \nThis is joint work with Carolyn Chun\, James Dylan Douthitt\, Wayne Ge\, Matthew E. Kroeker\, and Peter Nelson.
URL:https://dimag.ibs.re.kr/event/2025-08-05/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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