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:

• verify.sage – the verification Sage script
• 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 uD, 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

The file flagmatic-flags_pruned.rat was generated by Flagmatic 1.5.1 (flagmatic --n 7 --forbid-k4-), and then the auxiliary file flags.rat was pruned only to the six relevant types ι1, …, ι6. Note that in the Flagmatic output, they correspond to the types indexed by 7, 8, 9, 10, 11 and 14, respectively.

USAGE: run sage verify.sage

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