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Hypergraph Ramsey numbers
"... The Ramsey number rk(s, n) is the minimum N such that every redblue coloring of the ktuples of an Nelement set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all ktuples from this set are red (blue). In this paper we obtain new estimates for seve ..."
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The Ramsey number rk(s, n) is the minimum N such that every redblue coloring of the ktuples of an Nelement set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all ktuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for rk(s, n) for k ≥ 3 and s fixed. In particular, we show that r3(s, n) ≤ 2 ns−2 log n, which improves by a factor of n s−2 /polylogn the exponent of the previous upper bound of Erdős and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there are constants c1, c2> 0 such that c1 sn log(n/s) r3(s, n) ≥ 2 for all 4 ≤ s ≤ c2n. When s is a constant, it gives the first superexponential lower bound for r3(s, n), answering an open question posed by Erdős and Hajnal in 1972. Next, we consider the 3color Ramsey number r3(n, n, n), which is the minimum N such that every 3coloring of the triples of an Nelement set contains a monochromatic set of size n. Improving another old result of Erdős and Hajnal, we show that n nclog r3(n, n, n) ≥ 2. Finally, we make some progress on related hypergraph Ramseytype problems. 1
Graphs with many copies of a given subgraph
"... Let c> 0, and H be a fixed graph of order r. Every graph on n vertices containing at least cnr copies of H contains a “blowup ” of H with r − 1 vertex classes of size ⌊ cr2 ln n ⌋ and one vertex class of size greater than n1−cr−1. A similar result holds for induced copies of H. Main results Supp ..."
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Cited by 5 (4 self)
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Let c> 0, and H be a fixed graph of order r. Every graph on n vertices containing at least cnr copies of H contains a “blowup ” of H with r − 1 vertex classes of size ⌊ cr2 ln n ⌋ and one vertex class of size greater than n1−cr−1. A similar result holds for induced copies of H. Main results Suppose that a graph G of order n contains cnr copies of a given subgraph H on r vertices. How large “blowup ” of H must G contain? When H is an rclique, this question was answered in [3]: G contains a complete rpartite graph with r − 1 parts of size ⌊ cr ln n ⌋ and one part larger than n1−cr−1. The aim of this note is to answer this question for any subgraph H. First we define precisely a “blowup ” of a graph: given a graph H of order r and positive integers x1,..., xr, we write H(x1,..., xr) for the graph obtained by replacing each vertex u ∈ V (H) with a set Vu of size xu and each edge uv ∈ E(H) with a complete bipartite graph with vertex classes Vu and Vv.
Turán’s theorem inverted
, 2008
"... Let K + r (s1,...,sr) be the complete rpartite graph with parts of size s1 ≥ 2,s2,...,sr with an edge added to the first part. Letting tr (n) be the number of edges of the rpartite Turán graph of order n, we prove that: (A) For all r ≥ 2 and all sufficiently small ε> 0, every graph of sufficien ..."
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Let K + r (s1,...,sr) be the complete rpartite graph with parts of size s1 ≥ 2,s2,...,sr with an edge added to the first part. Letting tr (n) be the number of edges of the rpartite Turán graph of order n, we prove that: (A) For all r ≥ 2 and all sufficiently small ε> 0, every graph of sufficiently large order n with tr (n) + 1 edges contains a K + r ⌊cln n⌋,..., ⌊cln n⌋, n 1− √ c (B) For all r ≥ 2, there exists c> 0 such that every graph of sufficiently large order n with tr (n) + 1 edges contains a K + r (⌊cln n⌋,..., ⌊cln n⌋). These assertions extend results of Erdős from 1963. We also give corresponding stability results Keywords: clique; rpartite graph; stability, Turán’s theorem
unknown title
, 2009
"... In this note we strengthen the stability theorem of Erdos and Simonovits. Write Kr (s1; : : : ; sr) for the complete rpartite graph with classes of size s1; : : : ; sr and Tr (n) for the rpartite Turán graph of order n: Our main result is: For all r 2 and all su ¢ ciently small c> 0; "&g ..."
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In this note we strengthen the stability theorem of Erdos and Simonovits. Write Kr (s1; : : : ; sr) for the complete rpartite graph with classes of size s1; : : : ; sr and Tr (n) for the rpartite Turán graph of order n: Our main result is: For all r 2 and all su ¢ ciently small c> 0; "> 0; every graph G of su ¢ ciently large order n with e (G)> (1 1=r ")n2=2 satis
es one of the conditions: (a) G contains a Kr+1 bc lnnc; : : : ; bc lnnc;