Files needed for a verification of the proof of Lemma 2.8 in the paper "The codegree threshold of K4-" by Falgas-Ravry, Pikhurko, Vaughan 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 2.8 presented in the paper The codegree threshold of K4- authored by Victor Falgas-Ravry, Oleg Pikhurko, Emil Vaughan 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 2.8M)

  Description of the files in the archive:

  • lemma28.sage – the SAGE script verifying the identity (2.2)
  • flagmatic-flags_pruned.rat – the pairing densities for types ι₁, …, ι₆
  • ineq_codeg and ineq_tightpath – the coefficients of the CODEGREE and TIGHT-PATH expressions (expressed as a combination of 7-vertex 3-graphs)
  • solution-u and solution-c – the non-negative rationals u_D, and the positive rationals c₀, c₁ and c₂
  • solution-I_1, …, solution-I_6 – the rational matrices I₁, …, I₆
  • solution-Q_1.blockA, solution-Q_1.blockB, …, solution-Q_6.blockA, solution-Q_6.blockB – the positive definite matrices Q₁, …, Q₆, each decomposed into two diagonal blocks A and B
  • graph_lists/ – the listings of the flags involved in the proof
  • gen7.sage – the SAGE script (re)generating the file 'flagmatic_flags-pruned.rat' and the supplementary 'graph_lists/list6_iotaX' files
  • gen_flags.sage – the SAGE script (re)generating the file 'graph_lists/list7'
  • codeg.sage – the SAGE script (re)generating the file 'ineq_codeg' containing the coefficients of the 905 expressions D
  • tightpath.sage – the SAGE script (re)generating the file 'ineq_tightpath' containing the coefficients of the expressions P₀, P₁ and P₂
  • ext7.sage – the SAGE script computing the size of the intersection of the sets E_T and F₇
  • flags.sage – supplementary SAGE functions

  The file flagmatic-flags_pruned.rat can be also generated by Flagmatic 1.5.1 by running flagmatic --n 7 --forbid-k4-, and then pruning the auxiliary file flags.rat to the six relevant types ι₁, …, ι₆. Note that in Flagmatic, these correspond to SDP blocks indexed by 7, 8, 9, 10, 11 and 14, respectively.

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  USAGE: run sage lemma28.sage

  The verification requires around 2GB of memory and a 64bit version of Sage 9.x (tested with Sage 9.0, 9.2 and 9.5).