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Geometric constructions for Ramsey-Turan theory

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 نشر من قبل Katherine Staden
 تاريخ النشر 2021
  مجال البحث
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Combining two classical notions in extremal combinatorics, the study of Ramsey-Turan theory seeks to determine, for integers $mle n$ and $p leq q$, the number $mathsf{RT}_p(n,K_q,m)$, which is the maximum size of an $n$-vertex $K_q$-free graph in which every set of at least $m$ vertices contains a $K_p$. Two major open problems in this area from the 80s ask: (1) whether the asymptotic extremal structure for the general case exhibits certain periodic behaviour, resembling that of the special case when $p=2$; (2) constructing analogues of Bollobas-ErdH{o}s graphs with densities other than $1/2$. We refute the first conjecture by witnessing asymptotic extremal structures that are drastically different from the $p=2$ case, and address the second problem by constructing Bollobas-ErdH{o}s-type graphs using high dimensional complex spheres with all rational densities. Some matching upper bounds are also provided.



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