Let r be an integer, f (n) be a function, and H be a graph. Introduced by Erdős, Hajnal, Sós, and Szemerédi, the r-Ramsey-Turán number of H, RT r (n, H, f (n)), is defined to be the maximum number of edges in an n-vertex,In this note, using isoperimetric properties of the high-dimensional unit sphere, we construct graphs providing lower bounds for RT r (n, Kr+s, o(n)) for every 2 s r. These constructions are sharp for an infinite family of pairs of r and s. The only previous sharp construction (for such values of r and s) was by Bollobás and Erdős for r = s = 2.

Let t be an integer, f (n) a function, and H a graph. Define the t-Ramsey-Turán number of H, RT t (n, H, f (n)), to be the maximum number of edges in an n-vertex, H-free graph G with α t (G) ≤ f (n), where α t (G) is the maximum number of vertices in a K t -free induced subgraph of G. Erdős, Hajnal, Simonovits, Sós, and Szemerédi [5] posed several open questions about RT t (n, K s , o(n)), among them finding the minimum ℓ such that RT t (n, K t+ℓ , o(n)) = Ω(n 2 ), where it is easy to see that RT t (n, K t+1 , o(n)) = o(n 2 ). In this paper, we answer this question by proving that RT t (n, K t+2 , o(n)) = Ω(n 2 ); our constructions also imply several results on the Ramsey-Turán numbers of hypergraphs.

Let p(k) denote the partition function of k. For each k 2, we describe a list of p(k) − 1 quasirandom properties that a k-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness by Kohayakawa, Rödl, and Skokan, and by Conlon, Hàn, Person, and Schacht, and the spectral approach of Friedman and Wigderson. For each of the quasirandom properties that is described, we define the largest and the second largest eigenvalues. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of Conlon, Hàn, Person, and Schacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of Chung, Graham, and Wilson for graphs.

Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-GrahamWilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them.Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990's.

In this paper we consider spectral extremal problems for hypergraphs. We give two general criteria under which such results may be deduced from "strong stability" forms of the corresponding (pure) extremal results. These results hold for the α-spectral radius defined using the α-norm for any α > 1; the usual spectral radius is the case α = 2. Our results imply that any hypergraph Turán problem which has the stability property and whose extremal construction satisfies some rather mild continuity assumptions admits a corresponding spectral result. A particular example is to determine the maximum α-spectral radius of any 3-uniform hypergraph on n vertices not containing the Fano plane, when n is sufficiently large. Another is to determine the maximum α-spectral radius of any graph on n vertices not containing some fixed color-critical graph, when n is sufficiently large; this generalizes a theorem of Nikiforov who proved stronger results in the case α = 2. We also obtain an α-spectral version of the Erdős-Ko-Rado theorem on t-intersecting k-uniform hypergraphs.