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Applications of the Semi-Definite Method to the Turán Density Problem for 3-Graphs

Published online by Cambridge University Press:  07 December 2012

VICTOR FALGAS-RAVRY  and
EMIL R. VAUGHAN
VICTOR FALGAS-RAVRY
Affiliation:
Institutionen för matematik och matematisk statistik, Umeå Universitet, 901 87 Umeå, Sweden (e-mail: victor.falgas-ravry@math.umu.se)
EMIL R. VAUGHAN
Affiliation:
School of Electronic Engineering and Computer Science, Queen Mary, University of London, Mile End Road, London E1 4NS, UK (e-mail: emil79@gmail.com)
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Abstract

In this paper, we prove several new Turán density results for 3-graphs with independent neighbourhoods. We show:

\begin{align*} \pi (K_4^-, C_5, F_{3,2})=12/49, \pi (K_4^-, F_{3,2})=5/18 \textrm {and} \pi (J_4, F_{3,2})=\pi (J_5, F_{3,2})=3/8, \end{align*}
where Jt is the 3-graph consisting of a single vertex x together with a disjoint set A of size t and all $\binom{|A|}{2}$ 3-edges containing x. We also prove two Turán density results where we forbid certain induced subgraphs:
\begin{align*} \pi (F_{3,2}, \textrm { induced }K_4^-)=3/8 \textrm {and} \pi (K_5, 5\textrm {-set spanning exactly 8 edges})=3/4. \end{align*}
 The latter result is an analogue for K5 of Razborov's result that
\begin{align*} \pi (K_4, 4\textrm {-set spanning exactly 1 edge})=5/9. \end{align*}
We give several new constructions, conjectures and bounds for Turán densities of 3-graphs which should be of interest to researchers in the area. Our main tool is ‘Flagmatic’, an implementation of Razborov's semi-definite method, which we are making publicly available. In a bid to make the power of Razborov's method more widely accessible, we have tried to make Flagmatic as user-friendly as possible, hoping to remove thereby the major hurdle that needs to be cleared before using the semi-definite method. Finally, we spend some time reflecting on the limitations of our approach, and in particular on which problems we may be unable to solve. Our discussion of the ‘complexity barrier’ for the semi-definite method may be of general interest.

Keywords

Primary 05D05Secondary 05C65
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 1 , January 2013 , pp. 21 - 54
Copyright
Copyright © Cambridge University Press 2012

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