Pentagons vs. triangles

Bollobás, Béla; Gyri, Ervin

doi:10.1016/j.disc.2007.08.016

Cited by 99 publications

(75 citation statements)

References 4 publications

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“…A natural approach to the problem of finding a large independent set in a hypergraph with no given cycle is to combine Proposition with a bound on the Turán number of the cycle in question. This approach works when forbidding the family of Berge

l

‐cycles (a more general form of cycles than loose cycles; see Bollobás and Györi , Györi and Lemons for bounds on their Turán numbers), but fails for loose cycles because their Turán numbers are too large. The problem comes not only from

C_{l}^{r}

‐free constructions with small transversal number (eg, the three‐graph on

n

vertices with edges of the form

{v} \cup {a, b}

where

v

is fixed and

{a, b}

ranges over all pairs of vertices which do not include

v

), but also from constructions with large transversal number (eg, let

M

be a perfect graph matching of size

n

, and let

Z

be a set of

n

vertices disjoint from

M

; then the three‐graph with edge set

{e \cup {v} : e \in M, v \in Z}

C_{l}^{3}

‐free, and has transversal as well as independence number of linear size).…”

Section: Overviewmentioning

confidence: 99%

“…A natural approach to the problem of finding a large independent set in a hypergraph with no given cycle is to combine Proposition 2.2 with a bound on the Turán number of the cycle in question. This approach works when forbidding the family of Berge l-cycles (a more general form of cycles than loose cycles; see Bollobás and Györi [4], Györi and Lemons [13,14] for bounds on their Turán numbers), but fails for loose cycles because their Turán numbers are too large. The problem comes not only from C l r -free constructions with small transversal number (eg, the three-graph on n vertices with edges of the form…”

Section: Overviewmentioning

confidence: 99%

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The Ramsey number of loose cycles versus cliques

Méroueh

2018

Journal of Graph Theory

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Let R ( C l r , K n r ) be the Ramsey number of an r‐uniform loose cycle of length l versus an r‐uniform clique of order n. Kostochka et al. showed that for each fixed r ≥ 3, the order of magnitude of R ( C 3 r , K n r ) is n 3 ∕ 2 up to a polylogarithmic factor in n. They conjectured that for each r ≥ 3 we have R ( C 5 r , K n r ) = O ( n 5 ∕ 4 ). We prove that R ( C 5 3 , K n 3 ) = O ( n 4 ∕ 3 ), and more generally for every l ≥ 3 that R ( C l 3 , K n 3 ) = O ( n 1 + 1 ∕ ⌊ ( l + 1 ) ∕ 2 ⌋ ). We also prove that for every l ≥ 5 and r ≥ 4, R ( C l r , K n r ) = O ( n 1 + 1 ∕ ⌊ l ∕ 2 ⌋ ) if l is odd, which improves upon the result of Collier‐Cartaino et al. who proved that for every r ≥ 3 and l ≥ 4 we have R ( C l r , K n r ) = O ( n 1 + 1 ∕ ( ⌊ l ∕ 2 ⌋ − 1 ) ).

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l

C_{l}^{r}

‐free constructions with small transversal number (eg, the three‐graph on

n

vertices with edges of the form

{v} \cup {a, b}

where

v

is fixed and

{a, b}

ranges over all pairs of vertices which do not include

v

), but also from constructions with large transversal number (eg, let

M

be a perfect graph matching of size

n

, and let

Z

be a set of

n

vertices disjoint from

M

; then the three‐graph with edge set

{e \cup {v} : e \in M, v \in Z}

C_{l}^{3}

‐free, and has transversal as well as independence number of linear size).…”

Section: Overviewmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

The Ramsey number of loose cycles versus cliques

Méroueh

2018

Journal of Graph Theory

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“…Theorem 1.5 (Győri, Bollobás [2]). If G is a graph on n vertices containing no C 5 then the number of triangles in G is at most ( √ 2/4+ 1)n 3/2 + o(n 3/2 ).…”

Section: Theorem 14 (Győri Bollobás [2]) Let H Be a 3-uniform Hypementioning

confidence: 95%

3-uniform hypergraphs avoiding a given odd cycle

Győri

Lemons

2012

Combinatorica

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“…Motivated by a conjecture of Erdős on the maximum possible number of pentagons in a triangle‐free graph, Bollobás and Győri initiated the study of the natural converse of this problem. Let

e x (n, K_{3}, C_{5})

denote the maximum possible number of triangles in a graph on

n

vertices without containing a cycle of length five as a subgraph.…”

Section: Introductionmentioning

confidence: 99%

“…Let

e x (n, K_{3}, C_{5})

denote the maximum possible number of triangles in a graph on

n

vertices without containing a cycle of length five as a subgraph. Bollobás and Győri showed that

\frac{1}{3 \sqrt{3}} (1 + o (1)) n^{3 ∕ 2} \leq normale normalx (n, K_{3}, C_{5}) \leq \frac{5}{4} (1 + o (1)) n^{3 ∕ 2} .

…”

Section: Introductionmentioning

confidence: 99%

A note on the maximum number of triangles in a C₅‐free graph

Ergemlidze¹,

Methuku²,

Salia³

et al. 2018

Journal of Graph Theory

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Pentagons vs. triangles

Cited by 99 publications

References 4 publications

The Ramsey number of loose cycles versus cliques

The Ramsey number of loose cycles versus cliques

3-uniform hypergraphs avoiding a given odd cycle

A note on the maximum number of triangles in a C₅‐free graph

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Pentagons vs. triangles

Cited by 99 publications

References 4 publications

The Ramsey number of loose cycles versus cliques

The Ramsey number of loose cycles versus cliques

3-uniform hypergraphs avoiding a given odd cycle

A note on the maximum number of triangles in a C5‐free graph

Contact Info

Product

Resources

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A note on the maximum number of triangles in a C₅‐free graph