This web page contains data and verification scripts for the computer-assisted proofs used in my PhD thesis Analytic Methods in Combinatorics.

Files needed for a verification of the proof of Lemma 2.11 (Lemma 2.13 in the first draft):

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

  Description of the files in the archive:

  • lemma_2_13-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 the types σ_1, …, σ_8 (generated by flagmatic 1.5.1)
  • ineq1 and ineq2 – the inequalities (2.10) and (2.11) expressed as a combination of 7-vertex 3-graphs
  • gamma1 and gamma2 – coefficients for the inequalities (2.10) and (2.11)
  • vectors_sigma1, …, vectors_sigma8 – list of vectors v^j_i for matrices M_1, …, M_8
  • weights_sigma1, …, weights_sigma8 – weights w^j_i of vectors v^j_i for matrices M_1, …, M_8

  USAGE: run sage lemma_2_13-verify.sage

  • Note that the computation requires around 7GB of RAM memory and hence a 64bit version of Sage.
  • Also note that it takes around 60 minutes to run the computation (tested on Intel Core i7-3770 with 16GB RAM and AMD FX-8150 with 16GB RAM).
  
  
• Final vector generated by the script (gzip) Note the compressed size is 46M. This is file is not needed for the verification itself.
• list of 7-vertex 3-graphs and flags rooted on σ_1, …, σ_8 (gzip) Lists were generated by flagmatic 1.5.1. This file is not needed for the verification itself.

Files needed for a verification of the proof of Theorem 2.20 (Theorem 2.22 in the first draft):

• verification script and data (gzip)

  Description of the files in the archive:

  • theorem_2_22-verify.sage – the verification Sage script
  • pair1, …, pair4 – the pairing densities for the types σ_1, …, σ_4
  • M1, …, M4 – the positive definite matrices M'_1, …, M'_4
  • T1, …, T4 – the matrices T_1, … T_4

  USAGE: run sage theorem_2_22-verify.sage

• list of 6-vertex 3-graphs and flags rooted on σ_1, …, σ_4 (gzip) This file is not needed for the verification itself.

Files needed for a verification of the proof of Theorem 3.4:

• verification script and data (gzip)

  Description of the files in the archive:

  • thm_3_4-verify.sage – the verification Sage script
  • goal_vector – the vector of coefficients for 7-vertex graphs denoting K_4 density
  • pair1, pair2 – the pairing densities for the types σ_1 and σ_2
  • N1, N2 – the positive definite matrices N_1 and N_2
  • D1, D2 – the matrices D_1 and D_2

  USAGE: run sage thm_3_4-verify.sage

• list of 7-vertex permutation graphs and flags rooted on σ_1 and σ_2 (gzip) This file is not needed for the verification itself.

Files needed for a verification of the proof of Theorem 3.19:

• verification script and data (gzip)

  Description of the files in the archive:

  • thm_3_19-verify.sage – the verification Sage script
  • goal_vector – the vector of coefficients for 7-vertex graphs denoting K_5 density
  • pair1, …, pair4 – the pairing densities for the types σ_1, …, σ_4
  • N1, N2, M3, M4 – the positive definite matrices N_1, N_2, M_3 and M_4
  • D1, D2 – the matrices D_1 and D_2

  USAGE: run sage thm_3_19-verify.sage

• list of 7-vertex permutation graphs and flags rooted on σ_1, …, σ_4 (gzip) This file is not needed for the verification itself.

Files needed for a verification of the proof of Theorem 3.22:

• verification script and data (gzip) Note the compressed size of the archive is 5M.

  Description of the files in the archive:

  • thm_3_22-verify.sage – the verification Sage script
  • goal_vector – the vector of coefficients for 8-vertex graphs denoting K_6 density
  • pair1, …, pair31 – the pairing densities for the types σ_1, …, σ_31
  • N1, N2, N3, M4, N5, N6, M7, N8, M9, …, N31 – the positive definite matrices
  • D1, D2, D3, D5, D6, D8 – the matrices D_1, D_2, D_3, D_5, D_6 and D_8

  USAGE: run sage thm_3_22-verify.sage

  • Note that the computation requires around 3.5GB of RAM memory and hence a 64bit version of Sage.
  • Also note that it takes around 20 minutes to run the computation (tested on Intel Core i7-3770 with 16GB RAM).
  
  
• list of 8-vertex permutation graphs and flags rooted on σ_1, …, σ_31 (gzip) This file is not needed for the verification itself.