More on the extremal number of subdivisions

Conlon, David; Janzer, Oliver; Lee, Joonkyung

doi:10.48550/arxiv.1903.10631

Cited by 10 publications

(43 citation statements)

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“…Let L s,t (k) by obtained from K k s,t by adding an extra vertex joined to all vertices in the part of K s,t of size t. Confirming a conjecture of Kang, Kim, and Liu [22], Conlon, Janzer, and Lee [6] showed that there exists t 0 such that for all integers s, k ≥ 1, t ≥ t 0 , ex(n, L s,t (k)) = Θ(n 1+ s sk+1 ), and thus establishing 1+ s sk+1 as Turán exponents. Subsequently, in verifying a conjecture of Conlon, Janzer, and Lee [6], Janzer [16] proved that there exists a t 0 such that for all integers s, k ≥ 2, t ≥ t 0 , ex(n, K k s,t ) = Θ(n 1+ s−1 sk ), thus establishing 1 + s−1 sk as Turán exponents. Earlier, Conlon, Janzer, Lee [6] had proven the conjecture for k = 2, while Jiang and Qiu [20] proved the conjecture for k = 3, 4.…”

Section: Introduction 1rational Exponent Conjecturesupporting

confidence: 53%

Many Turan exponents via subdivisions

Jiang¹,

Yu²

2019

Preprint

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Given a graph H and a positive integer n, the Turán number ex(n, H) is the maximum number of edges in an n-vertex graph that does not contain H as a subgraph. A real number r ∈ (1, 2) is called a Turán exponent if there exists a bipartite graph H such that ex(n, H) = Θ(n r ). A long-standing conjecture of Erdős and Simonovits states that 1 + p q is a Turán exponent for all positive integers p and q with q > p. In this paper, we build on recent developments on the conjecture to establish a large family of new Turán exponents. In particular, it follows from our main result that 1 + p q is a Turán exponent for all positive integers p and q with q > p 2 .

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Section: Introduction 1rational Exponent Conjecturesupporting

confidence: 53%

Many Turan exponents via subdivisions

Jiang¹,

Yu²

2019

Preprint

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“…We give two corollaries in this fashion for the asymmetric version, that is, when t > s. The corollaries are obtained by choosing H as 1-subdivision of complete graphs and 1-subdivision of complete bipartite graphs, respectively. We utilise known extremal numbers of 1-subdivision of complete graphs ex(n, sub [7,17]).…”

Section: Theorem 12 ([8]mentioning

confidence: 99%

Color isomorphic even cycles and a related Ramsey problem

Zhang²,

Jing³

et al. 2020

Preprint

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In this paper, we first study a new extremal problem recently posed by Conlon and Tyomkyn (arXiv: 2002.00921). Given a graph H and an integer k 2, let f k (n, H) be the smallest number of colors c such that there exists a proper edge-coloring of the complete graph K n with c colors containing no k vertex-disjoint color-isomorphic copies of H. Using algebraic properties of polynomials over finite fields, we give an explicit edgecoloring of K n and show that f k (n, C 4 ) = Θ(n) when k 3. The methods we used in the edge-coloring may be of some independent interest. We also consider a related generalized Ramsey problem. For given graphs G and H, let r(G, H, q) be the minimum number of edge-colors (not necessarily proper) of G, such that the edges of every copy of H ⊆ G together receive at least q distinct colors. Establishing the relation to the Turán number of specified bipartite graphs, we obtain some general lower bounds for r(K n,n , K s,t , q) with a broad range of q.

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“…For more information on the recent active study of the Turán problem for subdivisions, we refer the readers to [6,7,12,13,14,15] and the references therein.…”

Section: Conjecture 12 ([8]mentioning

confidence: 99%

“…In this paper, we focus on the 1-subdivision of complete bipartite graphs. In [6], Conlon, Janzer and Lee showed that ex(n,…”

Section: Conjecture 12 ([8]mentioning

confidence: 99%

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On the Turán number of 1-subdivision of $K_{3,t}$

Zhang,

Xu,

2020

Preprint

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For a graph H, the 1-subdivision of H, denoted by H ′ , is the graph obtained by replacing the edges of H by internally disjoint paths of length 2. Recently, Conlon, Janzer and Lee (arXiv: 1903.10631) asked the following question: For any integer s ≥ 2, estimate the smallest t such that ex(n, K ′ s,t ) = Ω(n). In this paper, we consider the case s = 3. More precisely, we provide an explicit construction giving ex(n, K ′ 3,30 ) = Ω(n3 ), which reduces the estimation for the smallest value of t from a magnitude of 10 56 to the number 30. The construction is algebraic, which is based on some equations over finite fields.

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More on the extremal number of subdivisions

Cited by 10 publications

References 0 publications

Many Turan exponents via subdivisions

Many Turan exponents via subdivisions

Color isomorphic even cycles and a related Ramsey problem

On the Turán number of 1-subdivision of $K_{3,t}$

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