### Abstract

Our main result is a formulation and proof of the reverse hypercontractive inequality in the sum-of-squares (SOS) proof system. As a consequence we show that for any constant 0 < γ ≤ 1/4, the SOS/Lasserre SDP hierarchy at degree 4[1/4gamma;]certifies the statement "the maximum independent set in the Frankl-Rödl graph FR_{γ}^{n} has fractional size o(1)''.Here FR_{γ}^{n} = (V, E) is the graph with V = (0,1)^{n} and (x, y) ∈ E wheneverΔ(x, y) = (1-γ)n (an even integer). In particular, we show the degree-4 SOS algorithm certifies the chromatic number lower bound "Χ(FR^{n} _{1/4}) = ω(1)", relaxations cannot even certify "Χ(FR_{1/4}^{n} > 3^{n}. Finally, we also give an SOS proof of (a generalization of) the sharp (2, q)-hypercontractive inequality for any even integer q.

Original language | English (US) |
---|---|

Pages (from-to) | 1-21 |

Number of pages | 21 |

Journal | Discrete Analysis |

Volume | 4 |

Issue number | 2016 |

State | Published - Jan 1 2016 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Analysis*,

*4*(2016), 1-21.

**Hypercontractive inequalities via SOS, and the Frankl-Rödl graph.** / Kauers, Manuel; O'Donnell, Ryan; Tan, Li Yang; Zhou, Yuan.

Research output: Contribution to journal › Article

*Discrete Analysis*, vol. 4, no. 2016, pp. 1-21.

}

TY - JOUR

T1 - Hypercontractive inequalities via SOS, and the Frankl-Rödl graph

AU - Kauers, Manuel

AU - O'Donnell, Ryan

AU - Tan, Li Yang

AU - Zhou, Yuan

PY - 2016/1/1

Y1 - 2016/1/1

N2 - Our main result is a formulation and proof of the reverse hypercontractive inequality in the sum-of-squares (SOS) proof system. As a consequence we show that for any constant 0 < γ ≤ 1/4, the SOS/Lasserre SDP hierarchy at degree 4[1/4gamma;]certifies the statement "the maximum independent set in the Frankl-Rödl graph FRγn has fractional size o(1)''.Here FRγn = (V, E) is the graph with V = (0,1)n and (x, y) ∈ E wheneverΔ(x, y) = (1-γ)n (an even integer). In particular, we show the degree-4 SOS algorithm certifies the chromatic number lower bound "Χ(FRn 1/4) = ω(1)", relaxations cannot even certify "Χ(FR1/4n > 3n. Finally, we also give an SOS proof of (a generalization of) the sharp (2, q)-hypercontractive inequality for any even integer q.

AB - Our main result is a formulation and proof of the reverse hypercontractive inequality in the sum-of-squares (SOS) proof system. As a consequence we show that for any constant 0 < γ ≤ 1/4, the SOS/Lasserre SDP hierarchy at degree 4[1/4gamma;]certifies the statement "the maximum independent set in the Frankl-Rödl graph FRγn has fractional size o(1)''.Here FRγn = (V, E) is the graph with V = (0,1)n and (x, y) ∈ E wheneverΔ(x, y) = (1-γ)n (an even integer). In particular, we show the degree-4 SOS algorithm certifies the chromatic number lower bound "Χ(FRn 1/4) = ω(1)", relaxations cannot even certify "Χ(FR1/4n > 3n. Finally, we also give an SOS proof of (a generalization of) the sharp (2, q)-hypercontractive inequality for any even integer q.

UR - http://www.scopus.com/inward/record.url?scp=85038570800&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85038570800&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85038570800

VL - 4

SP - 1

EP - 21

JO - Discrete Analysis

JF - Discrete Analysis

SN - 2397-3129

IS - 2016

ER -