?ュ??瀛﹁??锛???楦挎????

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?ュ???堕?达?2024骞???18?ワ??ㄥ??锛?16:00-17:00

?ュ???扮?癸?瀛︽椿??灞?1005浼?璁??

?ュ????瑕?锛?We establish a novel connection between the well-known chromatic threshold problem in extremal combinatorics and the celebrated $(p,q)$-theorem in discrete geometry. In particular, for a graph $G$ with bounded clique number and a natural density condition, we prove a $(p,q)$-theorem for an abstract convexity space associated with $G$. Our result strengthens those of Thomassen and Nikiforov on the chromatic threshold of cliques. Our $(p,q)$-theorem can also be viewed as a $\chi$-boundedness result for (what we call) ultra maximal $K_r$-free graphs. 

We further show that the graphs under study are blow-ups of constant size graphs, improving a result of Oberkampf and Schacht on homomorphism threshold of cliques. Our result unravels the cause underpinning such a blow-up phenomenon, differentiating the chromatic and homomorphism threshold problems for cliques. Our result implies that for the homomorphism threshold problem, rather than the minimum degree condition usually considered in the literature, the decisive factor is a clique density condition on co-neighborhoods of vertices.

?ュ??浜虹?浠?锛? ??楦挎????2015骞村?ㄤ??╄?浼?澶у???宸寸撼-棣?妲?????UIUC)??寰???澹??浣?锛?甯?浠?J贸zsef Balogh??????2019骞村?ㄥ??濞?澶у?(Warwick U)??寰?缁?韬?????锛?骞舵???疯?卞?界??????帮?UKRI锛????ラ?琚?濂???2022骞村???ラ?╁?藉?虹?绉?瀛︾??绌堕??IBS)浠婚?甯??瀛﹀?锛??版???舵???煎??姒???缁?????绌剁?(ECOPRO)??棰?澶翠汉??2023骞磋捣锛???浠诲?捐?缁???棰???椤剁骇????Siam Journal on Discrete Mathematics??富缂?????绌堕??????????间?姒???缁??????捐???绂绘?ｅ??浣???Ramsey??璁恒??缁????拌?????ournal of the American Mathematical Society绛?????涓???琛ㄨ???50浣?绡???