Seminars on Discrete Mathematics: Chromatic, Homomorphism and Blowup thresholds
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：Xinqi Huang (University of Science Technology of China)
： 2025-12-30 15:00
：Conference Room S208 at Experiment Building at Haiyun Campus
Speaker：Xinqi Huang (University of Science and Technology of China)
Time：2025-12-30 15:00
Location：Conference Room S208 at Experiment Building at Haiyun Campus
Abstract:
In 1973, Erdős and Simonovits initiated the study of minimum-degree conditions forcing H-free graphs to have bounded chromatic number, leading to the notion of the chromatic threshold of H. This problem attracted sustained attention for decades and was completely determined by Allen, Böttcher, Griffiths, Kohayakawa, and Morris in 2013.
A natural strengthening asks whether one can force not only bounded chromatic number (equivalently, a bounded-order homomorphic image), but a bounded-order homomorphic image that is itself H-free; this leads to the much less understood theory of homomorphism thresholds, originating from questions raised by Thomassen in 2002.
Motivated by these problems, we recently introduced a new notion, the blow-up threshold, which strengthens homomorphism-type conclusions by requiring a dense maximal H-free graph to arise as a blow-up of a bounded-order template. In this talk, we survey the history of these thresholds and present our recent progress.
