Turán numbers for Berge-hypergraphs and related extremal problems

Palmer, Cory; Tait, Michael; Timmons, Craig; Wagner, Adam Zsolt

doi:10.1016/j.disc.2019.02.003

Cited by 35 publications

(28 citation statements)

References 28 publications

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“…Recently, Palmer et al considered extremal problems for Berge‐hypergraphs and proved our theorem for the case

ℓ = 4

. Note that for every

ℓ \geq 4

, we have

{Forb}_{L} (n, r, ℓ + 1) \subseteq {Forb}_{L} (n, r, ℓ)

.…”

Section: Introductionmentioning

confidence: 74%

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On the number of linear hypergraphs of large girth

Balogh

2019

Journal of Graph Theory

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An r‐uniform linear cycle of length ℓ, denoted by C ℓ r, is an r‐graph with edges e 1 , … , e ℓ such that for every i ∈ [ ℓ − 1 ] ,0.33em ∣ e i ∩ e i + 1 ∣ = 1 ,0.33em ∣ e ℓ ∩ e 1 ∣ = 1, and e i ∩ e j = ∅ for all other pairs { i , j } ,0.33em i ≠ j. For every r ≥ 3 and ℓ ≥ 4, we show that there exists a constant C depending on r and ℓ such that the number of linear r‐graphs of girth ℓ is at most 2 C n 1 + 1 ∕ ⌊ ℓ ∕ 2 ⌋. Furthermore, we extend the result for ℓ = 4, proving that there exists a constant C depending on r such that the number of linear r‐graphs without C 4 r is at most 2 C n 3 ∕ 2. The idea of the proof is to reduce the hypergraph enumeration problems to some graph enumeration problems, and then apply a variant of the graph container method, which may be of independent interest. We extend a breakthrough result of Kleitman and Winston on the number of C 4‐free graphs, proving that the number of graphs containing at most n 2 ∕ 32 log 6 n0.33em C 4's is at most 2 11 n 3 ∕ 2, for sufficiently large n. We further show that for every r ≥ 3 and ℓ ≥ 2, the number of graphs such that each of its edges is contained in only O ( 1 ) cycles of length at most 2 ℓ, is bounded by 2 3 ( ℓ + 1 ) n 1 + 1 ∕ ℓ asymptotically.

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“…Recently, Palmer et al considered extremal problems for Berge‐hypergraphs and proved our theorem for the case

ℓ = 4

. Note that for every

ℓ \geq 4

, we have

{Forb}_{L} (n, r, ℓ + 1) \subseteq {Forb}_{L} (n, r, ℓ)

.…”

Section: Introductionmentioning

confidence: 74%

“…for every i 2 ∈ , where the third inequality is given by Lemma 2.4. Combining (22) and (23), we obtain that the number of choices for S is…”

mentioning

confidence: 99%

On the number of linear hypergraphs of large girth

Balogh

2019

Journal of Graph Theory

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“…The general definition of a Berge-G for an arbitrary graph G was introduced by Gerbner and Palmer in [10]. Since their publication, the Turán problem for Berge-G-free hypergraphs has been investigated heavily (see, for example [2,22,11]). Complete graphs were considered in [17] (and recently [12]).…”

Section: Introductionmentioning

confidence: 99%

Ramsey Numbers of Berge-Hypergraphs and Related Structures

Salia¹,

Tompkins²,

Wang³

et al. 2019

Electron. J. Combin.

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For a graph G = (V, E), a hypergraph H is called a Berge-G, denoted by BG, if there exists an injection f : E(G) → E(H) such that for every e ∈ E(G), e ⊆ f (e). Let the Ramsey number R r (BG, BG) be the smallest integer n such that for any 2-edge-coloring of a complete r-uniform hypergraph on n vertices, there is a monochromatic Berge-G subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that R 3 (BK s , BK t ) = s + t − 3 for s, t ≥ 4 and max(s, t) ≥ 5 where BK n is a Berge-K n hypergraph. For higher uniformity, we show that R 4 (BK t , BK t ) = t + 1 for t ≥ 6 and R k (BK t , BK t ) = t for k ≥ 5 and t sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.

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“…Extremal problems for Berge hypergraphs have attracted the attention of many researchers, see e.g. [6,7,9,10,19,20,21,24].…”

Section: Introductionmentioning

confidence: 99%

Ramsey Problems for Berge Hypergraphs

Gerbner¹,

Methuku²,

Omidi³

et al. 2020

SIAM J. Discrete Math.

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For a graph G, a hypergraph H is a Berge copy of G (or a Berge-G in short), if there is a bijection f : E(G) → E(H) such that for each e ∈ E(G) we have e ⊆ f (e). We denote the family of r-uniform hypergraphs that are Berge copies of G by B r G.For families of r-uniform hypergraphs H and H ′ , we denote by R(H, H ′ ) the smallest number n such that in any red-blue coloring of the (hyper)edges of K r n (the complete r-uniform hypergraph on n vertices) there is a monochromatic blue copy of a hypergraph in H or a monochromatic red copy of a hypergraph in H ′ . R c (H) denotes the smallest number n such that in any coloring of the hyperedges of K r n with c colors, there is a monochromatic copy of a hypergraph in H.In this paper we initiate the general study of the Ramsey problem for Berge hypergraphs, and show that if r > 2c, then R c (B r K n ) = n. In the case r = 2c, we show that R c (B r K n ) = n + 1, and if G is a non-complete graph on n vertices, then R c (B r G) = n, assuming n is large enough. In the case r < 2c we also obtain bounds on R c (B r K n ). Moreover, we also determine the exact value of R(B 3 T 1 , B 3 T 2 ) for every pair of trees T

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Turán numbers for Berge-hypergraphs and related extremal problems

Cited by 35 publications

References 28 publications

On the number of linear hypergraphs of large girth

On the number of linear hypergraphs of large girth

Ramsey Numbers of Berge-Hypergraphs and Related Structures

Ramsey Problems for Berge Hypergraphs

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