On the Ramsey-Tur\'an density of triangles

One of the oldest results in modern graph theory, due to Mantel, asserts that every triangle-free graphs on $n$ vertices has at most $\lfloor n^2/4\rfloor$ edges. About half a century later Andr\'asfai studied dense triangle-free graphs and proved that the largest triangle-free graphs on $n$ vertices without independent sets of size $\alpha n$, where $2/5\le \alpha < 1/2$, are blow-ups of the pentagon. More than 50 further years have elapsed since Andr\'asfai's work. In this article we make the next step towards understanding the structure of dense triangle-free graphs without large independent sets. Notably, we determine the maximum size of triangle-free graphs~$G$ on $n$ vertices with $\alpha (G)\ge 3n/8$ and state a conjecture on the structure of the densest triangle-free graphs $G$ with $\alpha(G) > n/3$. We remark that the case $\alpha(G) \le n/3$ behaves differently, but due to the work of Brandt this situation is fairly well understood.Comment: Revised according to referee report

On Hamiltonian cycles in hypergraphs with dense link graphs

We show that every $k$-uniform hypergraph on $n$ vertices whose minimum $(k-2)$-degree is at least $(5/9+o(1))n^2/2$ contains a Hamiltonian cycle. A construction due to Han and Zhao shows that this minimum degree condition is optimal. The same result was proved independently by Lang and Sahueza-Matamala.Comment: Dedicated to Endre Szemer\'edi on the occasion of his 80th birthda

A hierarchy of maximal intersecting triple systems

We reach beyond the celebrated theorems of Erdȍs-Ko-Rado and Hilton-Milner, and a recent theorem of Han-Kohayakawa, and determine all maximal intersecting triples systems. It turns out that for each $n\geq 7$ there are exactly 15 pairwise non-isomorphic such systems (and 13 for $n=6$). We present our result in terms of a hierarchy of Turán numbers $\operatorname{ex}^{(s)}(n; M_2^{3})$, $s\geq 1$, where $M_2^{3}$ is a pair of disjoint triples. Moreover, owing to our unified approach, we provide short proofs of the above mentioned results (for triple systems only). The triangle $C_3$ is defined as $C_3=\{\{x_1,y_3,x_2\},\{x_1,y_2,x_3\},\{x_2,y_1,x_3\}\}$. Along the way we show that the largest intersecting triple system $H$ on $n\geq 6$ vertices, which is not a star and is triangle-free, consists of $\max\{10,n\}$ triples. This facilitates our main proof's philosophy which is to assume that $H$ contains a copy of the triangle and analyze how the remaining edges of $H$ intersect that copy