Files needed for a verification of the proof of Lemma 5 in the paper "A problem of Erdös and Sós on 3-graphs" by Glebov, Kráľ and Volec

This web-page contains the coefficients, the matrices and the script that can be used to verify the correctness of the proof of Lemma 5 presented in the paper A problem of Erdos and Sos on 3-graphs authored by Roman Glebov, Daniel Kráľ and Jan Volec. A preliminary version of the paper can be downloaded from arXiv.

• verification script and data (gzip; the compressed size of the archive is 13M)

  Description of the files in the archive:

  • verify.sage – the verification Sage script
  • extremal-7 – the densities of 7-vertex 3-graphs in the extremal limit (the 3-graphs are ordered as in the list of all K₄⁻-free graphs generated by flagmatic 1.5.1, see below)
  • flags.rat – the pairing densities for types σ_1, …, σ_8 (generated by flagmatic 1.5.1)
  • ineq1 and ineq2 – the inequalities (4) and (5) expressed as a combination of 7-vertex 3-graphs
  • gamma1 and gamma2 – coefficients for the inequalities (4) and (5)
  • vectors_sigma1, …, vectors_sigma8 – list of vectors v^j_i for matrices A_1, …, A_8
  • weights_sigma1, …, weights_sigma8 – weights w^j_i of vectors v^j_i for matrices A_1, …, A_8

  USAGE: run sage verify.sage

  • Note that the computation requires around 7GB of memory and a 64bit version of Sage 5.x (tested with Sage 5.3 and 5.4.1).
  • Also note that it takes between 45 and 60 minutes to run the computation (on Intel Core i7-3770 with 16GB RAM / AMD FX-8150 with 16GB RAM).

Additional related files (not needed for the verification itself):

• Final vector generated by the script (gzip; the compressed size is 46M)
• list of 7-vertex graphs and flags rooted on σ_1, …, σ_8 (generated by flagmatic 1.5.1)