PROTOCOLO DEL APRENDIZAJE DE 10 PALABRAS

H1 H2 H3 H4 H5 H6 H7 H8

H9 H10 H11

Figure 3.8: The set of graphs EXT5₇.

In this section, we follow the approach presented in Sections 3.2, 3.3 and 3.4 in order to fully characterize all sufficiently large 5-extremal permutations.

Let EXT5₇ be the set of all 7-vertex subgraphs up to a blind isomorphism that have a positive density in the conjectured extremal construction. It follows that EXT5₇={H₁, H2, . . . , H11}= EXT47∪ {H9, H10, H11}. The set EXT57 is depicted in

Figure 3.8. LetE₇5 :=F₇\EXT5₇. The main theorem of this section is the following.

Theorem 3.19. There exists a positive rational α such that the following is true. If(π)n∈Nis a convergent sequence of permutations andφ∈Hom+(A,R) is its limit,

then φ  K5−α· X H∈E5 7 H  ≥ 1 256.

Proof. As in the proof of Theorem 3.4, we let σ1 to be the 3-vertex type with no

edges andσ2 the 3-vertex type with the edgebc. Additionally, let σ3 and σ4 be two

specific types of order 5 given in Figure 3.9.

Again, we set bτ1,1 := bτ1,2 := abc, bτ2,1 := bτ2,4 := abc, and bτ2,2 := bτ2,3 :=

bca. The set T(σ3) contains 16 permutationsτ3,j. For each j∈[16], we definebτ3,j

in the following way:

• forτ3,1= 13542, set bτ3,1 :=abcde,

• forτ3,2= 14352, set bτ3,2 :=acdbe,

• forτ3,3= 15243, set bτ3,3 :=aebdc,

• forτ3,4= 15324, set bτ3,4 :=aedcb,

• forτ3,5= 24315, set bτ3,5 :=bcdea,

• forτ3,6= 24531, set bτ3,6 :=edcba,

• forτ3,7= 25341, set bτ3,7 :=ebdca,

• forτ3,8= 32415, set bτ3,8 :=cdbea,

• forτ3,9 = 34251, set bτ3,9 =cdbea,

• forτ3,10= 41325, set bτ3,10 =ebdca,

• forτ3,11= 42135, set bτ3,11 =edcba,

• forτ3,12= 42351, set bτ3,12 =bcdea,

• forτ3,13= 51342, set bτ3,13 =aedcb,

• forτ3,14= 51423, set bτ3,14 =aebdc,

• forτ3,15= 52314, set bτ3,15 =acdbe,

• forτ3,16= 53124, set bτ3,16 =abcde.

It holds that|Fσ3

6 |= 26 and |F7σ3|= 574. Similarly, the set T(σ4) has size 12. For

eachj ∈[12], we setbτ4,j as follows:

• forτ4,1= 14532, set bτ4,1 :=abcde,

• forτ4,2= 15342, set bτ4,2 :=adbce,

• forτ4,3= 15423, set bτ4,3 :=aedbc,

• forτ4,4= 23541, set bτ4,4 :=edcba,

• forτ4,5= 24351, set bτ4,5 :=dcbea,

• forτ4,6= 32451, set bτ4,6 :=cbdea,

• forτ4,7 = 34215, set bτ4,7 :=cbdea,

• forτ4,8 = 42315, set bτ4,8 :=dcbea,

• forτ4,9 = 43125, set bτ4,9 :=edcba,

• forτ4,10= 51243, set bτ4,10 :=aedbc,

• forτ4,11= 51324, set bτ4,11 :=adbce,

• forτ4,12= 52134, set bτ4,12 :=abcde.

In this case,|Fσ4

6 |= 28 and |F7σ4|= 624.

Based on the semidefinite method presented in Section 1.2, an instance of the SDP was used to find 4 symmetric positive semidefinite matricesM1, M2, M3

andM4 with rational entries such that for everyφ∈Hom+(A,R) we have

φ 4 X i=1 q xT_i Mixi y σi ! ≤φ  K5− 1 256−α· X H∈E5 7 H  , where

a

b

c

σ

a

b

c

σ

e

c

_d

b

a

σ

e

c

_d

b

a

σ

Figure 3.9: The typesσ1 through σ4 used in the proof of Theorem 3.19.

• the vector x1 ∈ (RF5σ1)|F σ₁

5 | is the vector whose j-th coordinate is equal to

thej-th element of the canonical base of RF₅σ1, • the vector x2 ∈ (RF₅σ2)|F

σ₂

5 | is the vector whose j-th coordinate is equal to

thej-th element of the canonical base of RF5σ2,

• the vector x3 ∈ (RF6σ3)|F σ3

6 | is the vector whose j-th coordinate is equal to

thej-th element of the canonical base of RF6σ3,

• the vector x4 ∈ (RF6σ4)|F σ4

6 | is the vector whose j-th coordinate is equal to

thej-th element of the canonical base of RF6σ4, and

• α= 22961176619/6306641510400 ∼ 0.00364.

The left-hand side of the inequality above is non-negative by (1.2).

As in the case of Theorem 3.4, the numerical values of the entries of the ma- trices M1, M2, M3 and M4 can be downloaded from the web page http://honza. ucw.cz/phd/. We also created a sage script called “thm 3 19-verify.sage”, which can be used to verify the computations.

The methods presented in Section 3.3 can be straightforwardly adopted also to the case of monotone subsequences of length 5. Specifically, they yield the fol- lowing stability result, which is an analogue of Theorem 3.9.

Theorem 3.20. For every εSTAB > 0 there exist δSTAB > 0 and nSTAB such that

the following is true. If G is a permutation graph on nSTAB ≥ n0 vertices with

f5(G) ≤ ₂₅₆1 +δSTAB, then G is isomorphic to either T4(n) or T4(n) after adding

The reasoning is essentially the same as for Theorem 3.9. For the sake of complete- ness, we give a proof in Appendix B.1.

The stability result together with Theorem 3.14 and Theorem 3.2 gives a complete characterization of 5-extremal permutations.

Theorem 3.21. There exists an integer n0 such that for every permutationτ ∈Sn,

where n≥n0, we have F5(τ)≥F5(τ5(n)). Furthermore, if F5(τ) =F5(τ5(n)), then

τ ∈ W5(n).