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Homomorphism and VC-dimension thresholds: spectra and separations
Lior Gishboliner, Xinqi Huang, Hong Liu
Minimum-degree thresholds ask when excluding a fixed graph forces a dense graph to admit a simple global description. For each fixed chromatic number, the chromatic threshold h…
Exact Homomorphism Thresholds Beyond Cliques
Xinqi Huang, Mingyuan Rong, Chong Shangguan
The chromatic threshold, originating in a question of Erdős and Simonovits, asks when a linear minimum-degree condition forces bounded chromatic number in H-free graphs. Motivated…
On the spectrum and structure of blowup thresholds
Xinqi Huang, Hong Liu, Mingyuan Rong
The chromatic threshold of Erdős and Simonovits asks when a minimum-degree condition forces every \(H\)-free graph to have bounded chromatic number. Thomassen's homomorphism thresh…
Colour diversity in spanning structures under Dirac-type conditions
Xinbu Cheng, Xinqi Huang, Hong Liu +2
Finding spanning structures with many distinct colours in properly edge-coloured graphs is a central theme in extremal combinatorics. A classical result of Andersen shows that ever…
Largest dyadic dual VC-dimension of non-piercing families
Xinqi Huang, Yuzhen Qi, Mingyuan Rong +1
The dyadic dual VC-dimension of a set system \( \mathcal{F} \) is the largest integer \( \ell \) such that there exist \( \ell \) sets \( F_1, F_{2}, \dots, F_\ell \in \mathcal{F}…
Interpolating chromatic and homomorphism thresholds
Xinqi Huang, Hong Liu, Mingyuan Rong +1
The problem of chromatic thresholds seeks for minimum degree conditions that ensure -free graphs to have a bounded chromatic number, or equivalently a bounded size homomorphic i…