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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, RTt(n,H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G with αt(G) ≤ f(n), where αt(G) is the maximum number of vertices in a Kt-free induced subgraph of G. Erdős, Hajnal, Simonovits, Sós, and Szemerédi [5] posed several open questions about… (More)

- József Balogh, John Lenz
- 2011

Let r be an integer, f(n) a function, and H a graph. Introduced by Erdős, Hajnal, Sós, and Szemerédi [8], the r-Ramsey-Turán number of H , RTr(n,H, f(n)), is defined to be the maximum number of edges in an n-vertex, H-free graph G with αr(G) ≤ f(n) where αr(G) denotes the Kr-independence number of G. In this note, using isoperimetric properties of the high… (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 by Kohayakawa, Rödl, and Skokan, and by Conlon, Hàn, Person, and Schacht, and the spectral approach of Friedman and Wigderson. 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)

- John Lenz, Dhruv Mubayi
- Journal of Graph Theory
- 2014

Let H1, . . . ,Hk be graphs. The multicolor Ramsey number r(H1, . . . ,Hk) is the minimum integer r such that in every edge-coloring of Kr by k colors, there is a monochromatic copy of Hi in color i for some 1 ≤ i ≤ k. In this paper, we investigate the multicolor Ramsey number r(K2,t, . . . ,K2,t,Km), determining the asymptotic behavior up to a… (More)

- John Lenz, Dhruv Mubayi
- Random Struct. Algorithms
- 2015

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 hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus… (More)

- József Balogh, John Lenz, Hehui Wu
- Discussiones Mathematicae Graph Theory
- 2011

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 (τα(G)/2)⌉)h(G) ≥ |V (G)| where τ ≈ 6.83. For graphs with α(G) ≥ 14, this improves on a recent result of Kawarabayashi… (More)

- Joshua N. Cooper, John Lenz, Timothy D. LeSaulnier, Paul S. Wenger, Douglas B. West
- Graphs and Combinatorics
- 2012

For a fixed graph H, a graph G is uniquely H-saturated if G does not contain H, but the addition of any edge from G to G completes exactly one copy of H. Using a combination of algebraic methods and counting arguments, we determine all the uniquely C4-saturated graphs; there are only ten of them.

- John Lenz, Dhruv Mubayi
- Discrete Mathematics
- 2017

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)

- Peter Keevash, John Lenz, Dhruv Mubayi
- SIAM J. Discrete Math.
- 2014

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)