Results 1  10
of
20
Weak quasirandomness for uniform hypergraphs
, 2009
"... We study quasirandom properties of kuniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known ChungGrahamWilson theorem fo ..."
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We study quasirandom properties of kuniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known ChungGrahamWilson theorem for quasirandom graphs. Moreover, let Kk be the complete graph on k vertices and M(k) the line graph of the graph of the kdimensional hypercube. We will show that the pair of graphs (Kk, M(k)) has the property that if the number of copies of both Kk and M(k) in another graph G are as expected in the random graph of density d, then G is quasirandom (in the sense of the ChungGrahamWilson theorem) with density close to d.
A STRUCTURAL RESULT FOR HYPERGRAPHS WITH MANY RESTRICTED EDGE COLORINGS
, 2010
"... For kuniform hypergraphs F and H and an integer r ≥ 2, let cr,F (H) denote the number of rcolorings of the set of hyperedges of H with no monochromatic copy of F and let cr,F (n) = maxH∈Hn cr,F (H), where the maximum runs over the family Hn of all kuniform hypergraphs on n vertices. Moreover, l ..."
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For kuniform hypergraphs F and H and an integer r ≥ 2, let cr,F (H) denote the number of rcolorings of the set of hyperedges of H with no monochromatic copy of F and let cr,F (n) = maxH∈Hn cr,F (H), where the maximum runs over the family Hn of all kuniform hypergraphs on n vertices. Moreover, let ex(n, F) be the usual Turán function, i.e., the maximum number of hyperedges of an nvertex kuniform hypergraph which contains no copy of F. In this paper, we consider the question for determining cr,F (n) for arbitrary fixed hypergraphs F and show cr,F (n) = r ex(n,F)+o(nk) for r = 2, 3. Moreover, we obtain a structural result for r = 2, 3 and any H with cr,F (H) ≥ rex(V (H),F) under the assumption that a stability result for the kuniform hypergraph F exists and V (H)  is sufficiently large. We also obtain exact results for cr,F (n) when F is a 3 or 4uniform generalized triangle and r = 2, 3, while cr,F (n) ≫ rex(n,F) for r ≥ 4 and n sufficiently large.
The QuasiRandomness of Hypergraph Cut Properties
"... Let α1,..., αk satisfy ∑ i αi = 1 and suppose a kuniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A1,..., Ak of sizes α1n,..., αkn, the number of edges intersecting A1,..., Ak is (asymptotically) the number one would expect to find in a ..."
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Cited by 5 (0 self)
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Let α1,..., αk satisfy ∑ i αi = 1 and suppose a kuniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A1,..., Ak of sizes α1n,..., αkn, the number of edges intersecting A1,..., Ak is (asymptotically) the number one would expect to find in a random kuniform hypergraph. Can we then infer that H is quasirandom? We show that the answer is negative if and only if α1 = · · · = αk = 1/k. This resolves an open problem raised in 1991 by Chung and Graham [J. AMS ’91]. While hypergraphs satisfying the property corresponding to α1 = · · · = αk = 1/k are not necessarily quasirandom, we manage to find a characterization of the hypergraphs satisfying this property. Somewhat surprisingly, it turns out that (essentially) there is a unique non quasirandom hypergraph satisfying this property. The proofs combine probabilistic and algebraic arguments with results from the theory of association schemes.
Eigenvalues of nonregular linearquasirandom hypergraphs
, 2014
"... 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 kuniform hypergraphs with loops. However, for k ≥ 3 no kuniform hypergraph is coregula ..."
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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 kuniform hypergraphs with loops. However, for k ≥ 3 no kuniform hypergraph is coregular. In this paper we remove the coregular requirement. Consequently, the characterization can be applied to kuniform hypergraphs; for example it is used in [19] to show that a construction of a kuniform hypergraph sequence is quasirandom.
Eigenvalues and Linear Quasirandom Hypergraphs
, 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 kuniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness of KohayakawaRödlSkokan and ConlonHànPersonSchacht and the spectral a ..."
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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 kuniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness of KohayakawaRödlSkokan and ConlonHànPersonSchacht and the spectral approach of FriedmanWigderson. For each of the quasirandom properties that are described, we define a largest and second largest eigenvalue. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of ConlonHànPersonSchacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of ChungGrahamWilson for graphs.
ON COLORINGS OF HYPERGRAPHS WITHOUT MONOCHROMATIC FANO PLANES
, 2009
"... For kuniform hypergraphs F and H and an integer r ≥ 2, let cr,F (H) denote the number of rcolorings of the set of hyperedges of H with no monochromatic copy of F and let cr,F (n) = maxH∈Hn cr,F (H), where the maximum runs over all kuniform hypergraphs on n vertices. Moreover, let ex(n, F) be th ..."
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Cited by 5 (4 self)
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For kuniform hypergraphs F and H and an integer r ≥ 2, let cr,F (H) denote the number of rcolorings of the set of hyperedges of H with no monochromatic copy of F and let cr,F (n) = maxH∈Hn cr,F (H), where the maximum runs over all kuniform hypergraphs on n vertices. Moreover, let ex(n, F) be the usual extremal or Turán function, i.e., the maximum number of hyperedges of an nvertex kuniform hypergraph which contains no copy of F. For complete graphs F = Kℓ and r = 2 Erdős and Rothschild conjectured that c2,K ℓ (n) = 2 ex(n,K ℓ). This conjecture was proved by Yuster for ℓ = 3 and by Alon, Balogh, Keevash, and Sudakov for arbitrary ℓ. In this paper, we consider the question for hypergraphs and show that in the special case, when F is the Fano plane and r = 2 or 3, then cr,F (n) = r ex(n,F) , while cr,F (n) ≫ r ex(n,F) for r ≥ 4.
Exact results on the number of restricted edge colorings for some families of linear hypergraphs
"... Abstract. For kuniform hypergraphs F and H and an integer r ≥ 2, let cr,F (H) denote the number of rcolorings of the set of hyperedges of H with no monochromatic copy of F and let cr,F (n) = maxH∈Hn cr,F (H), where the maximum is over the family Hn of all kuniform hypergraphs on n vertices. More ..."
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Abstract. For kuniform hypergraphs F and H and an integer r ≥ 2, let cr,F (H) denote the number of rcolorings of the set of hyperedges of H with no monochromatic copy of F and let cr,F (n) = maxH∈Hn cr,F (H), where the maximum is over the family Hn of all kuniform hypergraphs on n vertices. Moreover, let ex(n, F) be the usual extremal function, i.e., the maximum number of hyperedges of an nvertex kuniform hypergraph which contains no copy of F. Here, we consider the question for determining cr,F (n) for F being the kuniform expanded, complete 2graph Hk`+1 or the kuniform Fan(k)hypergraph Fk`+1 with core of size ( ` + 1), where ` ≥ k ≥ 3, and we show cr,F (n) = r ex(n,F) for r = 2, 3 and n large enough. Moreover, for r = 2 or r = 3, for kuniform hypergraphs H on n vertices the equality cr,F (H) = r ex(n,F) only holds if H is isomorphic to the `partite, kuniform Turán hypergraph on n vertices, once n is large enough. On the other hand, we show that cr,F (n) is exponentially larger than rex(n,F), if r ≥ 4. 1. Introduction and
Almost all cancellative triple systems are tripartite
, 2009
"... A triple system is cancellative if no three of its distinct edges satisfy A ∪ B = A ∪ C. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is cancellative. We prove that almost ..."
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A triple system is cancellative if no three of its distinct edges satisfy A ∪ B = A ∪ C. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is cancellative. We prove that almost all cancellative triple systems with vertex set [n] are tripartite. This sharpens a theorem of Nagle and Rödl [15] on the number of cancellative triple systems. It also extends recent work of Person and Schacht [16] who proved a similar result for triple systems without the Fano configuration. Our proof uses the hypergraph regularity lemma of Frankl and Rödl [11], and a stability theorem for cancellative triple systems due to Keevash and the second author [12]. 1