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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.
For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $rin mathbb{N}$, the $r$-independence number of $G$, denoted $alpha_r(G)$ is the largest size of a $K_r$-free se
We study Turan and Ramsey-type problems on edge-colored graphs. An edge-colored graph is called {em $varepsilon$-balanced} if each color class contains at least an $varepsilon$-proportion of its edges. Given a family $mathcal{F}$ of edge-colored grap
In this paper we study Turan and Ramsey numbers in linear triple systems, defined as $3$-uniform hypergraphs in which any two triples intersect in at most one vertex. A famous result of Ruzsa and Szemeredi is that for any fixed $c>0$ and large enou
We call a $4$-cycle in $K_{n_{1}, n_{2}, n_{3}}$ multipartite, denoted by $C_{4}^{text{multi}}$, if it contains at least one vertex in each part of $K_{n_{1}, n_{2}, n_{3}}$. The Turan number $text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{text{multi}})$ $bi
Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges ${a,b,c}, {c,d,e},$ and ${e,f,g}$. It is known that the $r$-color Ramsey number for $P$ is $R(P;r)=r+6$ for $rle 9$. The proof of this result relies on a caref