The Brown--Erd\H{o}s--S\'os problem is a fundamental problem in sparse hypergraph Tur\'an theory. For integers $r,k\ge 2$ and $s\ge r$, the problem asks for the maximum number $f^{(r)}(n;s,k)$ of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ distinct edges spanning at most $s$ vertices. In 1971, Brown...
Ting-Wei Chao, Xin-Qi Huang, Hong Liu· 0 citations
The chromatic threshold, originating in a question of Erd\H{o}s and Simonovits, asks when a linear minimum-degree condition forces bounded chromatic number in H-free graphs. Motivated by a question of Thomassen, the homomorphism threshold asks for the stronger conclusion that every such graph admits a homomorphism to a...
Xin-Qi Huang, Mingyuan Rong, C. Shangguan· 1 citation
Minimum-degree thresholds ask when excluding a fixed graph $H$ forces a dense graph to admit a simple global description. For each fixed chromatic number, the chromatic threshold has only three possible values. We show that this finite-spectrum phenomenon is special to chromatic threshold: already among $3$-chromatic g...
Lior Gishboliner, Xin-Qi Huang, Hong Liu· 1 citation
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