John Lenz

The poset of hypergraph quasirandomness

Chung and Graham began the systematic study of k-uniform 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 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.

Eigenvalues of non-regular linear quasirandom hypergraphs

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 k-uniform hypergraphs, but only for the so-called coregular k-uniform hypergraphs. In this paper, we extend this characterization to all k-uniform hypergraphs, not just the coregular ones. Specifically, we prove that if a k-uniform hypergraph satisfies the correct count of a specially defined four-cycle, then there is a gap between its first and second largest eigenvalue.

On the Ramsey-Turán numbers of graphs and hypergraphs

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 [6] posed several open questions about RTt(n,Ks, o(n)), among them finding the minimum ℓ such that RTt(n,Kt+ℓ, o(n)) = Ω(n2), where it is easy to see that RTt(n,Kt+1, o(n)) = o(n2). In this paper, we answer this question by proving that RTt(n,Kt+2, o(n)) = Ω(n2); our constructions also imply several results on the Ramsey-Turán numbers of hypergraphs.

Some exact Ramsey-Turán numbers

Let r be an integer, f(n) a function, and H a graph. Introduced by Erdýos, Hajnal, Sos, and Szemeredi (8), the r-Ramsey-Turan 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 dimensional unit sphere, we construct graphs providing lower bounds for RTr(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 was by Bollobas and Erdýos (6) for r = s = 2.

On the Chromatic Thresholds of Hypergraphs

Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the inmum of all non-negative reals c such that the subfamily of F comprising hy- pergraphs H with minimum degree at least c jV (H)j 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 hypergraphs. Luczak and Thomasse recently proved that the chromatic threshold of near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turan number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few innite families of nondegenerate hypergraphs whose Turan number is determined exactly. In an attempt to generalize Thomassens result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing fabc,abd,cdeg, the so-called generalized triangle. In order to prove upper bounds we introduce the concept of fiber bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fiber bundle dimension, a structural property ofber bundles which is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a generalized Kneser hypergraph. Using methods from extremal set theory, we prove that these generalized Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemeredi for graphs and might be of independent interest. Many open problems remain.

Multicolor Ramsey Numbers For Complete Bipartite Versus Complete Graphs

Let H1,...,Hk be graphs. The multicolor Ramsey number rH1,...,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 1i¾?ii¾?k. In this paper, we investigate the multicolor Ramsey number rK2,t,...,K2,t,Km, determining the asymptotic behavior up to a polylogarithmic factor for almost all ranges of t and m. Several different constructions are used for the lower bounds, including the random graph and explicit graphs built from finite fields. A technique of Alon and Rodl using the probabilistic method and spectral arguments is employed to supply tight lower bounds. A sample result isc1m2tlog4mti¾?rK2,t,K2,t,Kmi¾?c2m2tlog2mfor any t and m, where c1 and c2 are absolute constants.

Uniquely C 4 -Saturated Graphs

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

Hamilton cycles in quasirandom hypergraphs

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