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- József Balogh, John Lenz
- 2011

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 ver-tices in a K t-free induced subgraph of G. Erd˝ os, Hajnal, Simonovits, Sós, and Sze-merédi [5] posed several open… (More)

- John Lenz, Dhruv Mubayi
- 2013

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 of Kohayakawa-Rödl-Skokan and Conlon-H` an-Person-Schacht and the spectral approach of Friedman-Wigderson. For each of the quasirandom… (More)

Chung and Graham began the systematic study of hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on hyper-graph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus… (More)

- József Balogh, John Lenz
- 2011

Let r be an integer, f (n) a function, and H a graph. (n)), is defined to be the maximum number of edges in an n-vertex, H-free graph G with α r (G) ≤ f (n) where α r (G) denotes the K r-independence number of G. In this note, using isoperimetric properties of the high dimensional unit sphere, we construct graphs providing lower bounds for RT r (n, K r+s ,… (More)

The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger's Conjecture states that h(G) ≥ χ(G). Since χ(G)α(G) ≥ |V (G)|, Hadwiger's Conjecture implies that α(G)h(G) ≥ |V (G)|. We show that (2α(G) − ⌈log τ (τ α(G)/2)⌉)h(G) ≥ |V (G)| where τ ≈ 6.83. For graphs with α(G) ≥ 14, this improves on a recent result of… (More)

Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to coregular k-uniform hypergraphs with loops. However, for k ≥ 3 no k-uniform hypergraph is coregular. In this paper we remove the coregular requirement.… (More)

is the minimum integer r such that in every edge-coloring of K r by k colors, there is a monochromatic copy of H i in color i for some 1 ≤ i ≤ k. In this paper, we investigate the multicolor Ramsey number r(K 2,t ,. .. , K 2,t , K m), determining the asymptotic behavior up to a polylogarithmic factor for almost all ranges of t and m. Several different… (More)

Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hy-pergraphs H with minimum degree at least c |V (H)| r−1 has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for… (More)

- John Lenz, Dhruv Mubayi
- 2012

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 hypergraph quasirandomness, beginning with the early work of Chung and Graham and Frankl-Rödl related to strong hypergraph regularity, the spectral approach of… (More)

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… (More)