Lower bounds of Ramsey numbers based on cubic residues

Su, Wenlong; Li, Qiao; Luo, Haipeng; Li, Guiqing

doi:10.1016/s0012-365x(01)00283-7

Cited by 12 publications

(5 citation statements)

References 7 publications

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“…First, if m | (n − 1) then, rather than quadratic residues in F n , one may determine adjacency by means of the multiplicative subgroup of m-th powers and its cosets. Having similar quasirandom properties [2,27], such graphs proved useful in Ramsey theory [15,24,42]. They appear also in the classification of graphs with strong symmetries [32,37].…”

Section: Explicit Construction Of Ample Complexesmentioning

confidence: 99%

Ample simplicial complexes

Even-Zohar

Farber

Mead

2022

European Journal of Mathematics

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Motivated by potential applications in network theory, engineering and computer science, we study r-ample simplicial complexes. These complexes can be viewed as finite approximations to the Rado complex which has a remarkable property of indestructibility, in the sense that removing any finite number of its simplexes leaves a complex isomorphic to itself. We prove that an r-ample simplicial complex is simply connected and 2-connected for r large. The number n of vertexes of an r-ample simplicial complex satisfies $$\exp \bigl (\Omega \bigl (\frac{2^r}{\sqrt{r}}\bigr )\bigr )$$ exp ( Ω ( 2 r r ) ) . We use the probabilistic method to establish the existence of r-ample simplicial complexes with n vertexes for any $$n>r 2^r 2^{2^r}$$ n > r 2 r 2 2 r . Finally, we introduce the iterated Paley simplicial complexes, which are explicitly constructed r-ample simplicial complexes with nearly optimal number of vertexes.

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Section: Explicit Construction Of Ample Complexesmentioning

confidence: 99%

Ample simplicial complexes

Even-Zohar

Farber

Mead

2022

European Journal of Mathematics

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“…First, if m|(n − 1) then, rather than quadratic residues in F n , one may determine adjacency by means of the multiplicative subgroup of mth powers and its cosets. Having similar quasirandom properties [KP04, AC06], such graphs proved useful in Ramsey theory [Cla79,GT83,SLLL02]. They appear also in the classification of graphs with strong symmetries [Pei01,LLP09].…”

Section: Explicit Construction Of Ample Complexesmentioning

confidence: 99%

Ample simplicial complexes

Even-Zohar¹,

Farber²,

Mead³

2020

Preprint

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Motivated by potential applications in network theory, engineering and computer science, we study r-ample simplicial complexes. These complexes can be viewed as finite approximations to the Rado complex which has a remarkable property of indestructibility, in the sense that removing any finite number of its simplexes leaves a complex isomorphic to itself. We prove that an r-ample simplicial complex is simply connected and 2-connected for r large. The number n of vertexes of an r-ample simplicial complex satisfies exp(Ω( 2 r √ r )). We use the probabilistic method to establish the existence of r-ample simplicial complexes with n vertexes for any n > r2 r 2 2 r . Finally, we introduce the iterated Paley simplicial complexes, which are explicitly constructed r-ample simplicial complexes with nearly optimal number of vertexes.

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“…50} into a disjoint union of sets S 1 and S 2 consisting of quadratic residues and non-residues modulo 101, respectively, and argue that neither of the circulant graphs G 101 (S 1 ) and G 101 (S 2 ) contains K 6 as a subgraph. More recently, Su et al [4] utilized cubic residues in a similar fashion, sharpening the best known lower bounds on multiple Ramsey numbers.…”

Section: Using Quartic Residues To Improve Lower Boundsmentioning

confidence: 99%

Improved Lower Bounds on the Classical Ramsey Numbers R(4,22) and R(4,25)

Lindsay,

Cain

2015

Preprint

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Circulant graphs have been used to effectively establish lower bounds on many classical Ramsey numbers. Here, we construct circulant graphs of prime order that sharpen the best published lower bounds on several Ramsey numbers. Generalizing previous work in which quadratic and cubic residues were used to construct circulant graphs for the same purpose, we report on the use of quartic and higher-order residues.

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Lower bounds of Ramsey numbers based on cubic residues

Cited by 12 publications

References 7 publications

Ample simplicial complexes

Ample simplicial complexes

Ample simplicial complexes

Improved Lower Bounds on the Classical Ramsey Numbers R(4,22) and R(4,25)

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