Read's conjecture

{{Short description|Mathematical theorem first conjectured by Ronald Read}}

Read's conjecture is a conjecture, first made by Ronald Read, about the unimodality of the coefficients of chromatic polynomials in the context of graph theory.{{Cite journal |last=Baker |first=Matthew |date=January 2018 |title=Hodge theory in combinatorics |url=https://www.ams.org/bull/2018-55-01/S0273-0979-2017-01599-6/ |journal=Bulletin of the American Mathematical Society |language=en |volume=55 |issue=1 |pages=57–80 |doi=10.1090/bull/1599 |s2cid=51813455 |issn=0273-0979|doi-access=free |arxiv=1705.07960 }}R. C. Read, An introduction to chromatic polynomials, J. Combinatorial Theory 4 (1968), 52–71. MR0224505 (37:104) In 1974, S. G. Hoggar tightened this to the conjecture that the coefficients must be strongly log-concave. Hoggar's version of the conjecture is called the Read–Hoggar conjecture.{{Cite journal |last=Hoggar |first=S. G |date=1974-06-01 |title=Chromatic polynomials and logarithmic concavity |journal=Journal of Combinatorial Theory|series=Series B|language=en |volume=16 |issue=3 |pages=248–254 |doi=10.1016/0095-8956(74)90071-9 |doi-access=free |issn=0095-8956}}{{Cite web|title=Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries|url=https://web.northeastern.edu/martsinkovsky/p/rtrt/20152016/huh-slides.pdf|first=June|last=Huh}}

The Read–Hoggar conjecture had been unresolved for more than 40 years before June Huh proved it in 2009, during his PhD studies, using methods from algebraic geometry.{{Cite magazine|date=5 July 2022|title=He Dropped Out to Become a Poet. Now He's Won a Fields Medal.|url=https://www.quantamagazine.org/june-huh-high-school-dropout-wins-the-fields-medal-20220705|access-date=5 July 2022|magazine=Quanta Magazine|language=en}}{{cite journal|last=Kalai |first= Gil |title=The Work of June Huh| journal=Proceedings of the International Congress of Mathematicians 2022 |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2022/laudatio-jh.pdf |date=July 2022 |pages=1–16}}, pp. 2–4.{{cite journal |last=Huh |first=June | authorlink=June Huh |title=Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs |arxiv=1008.4749 |journal=Journal of the American Mathematical Society | volume=25 | date=2012 |issue=3 | pages=907–927 | doi=10.1090/S0894-0347-2012-00731-0 | doi-access=free}}