Decompositions

In this section, we introduce a grid-like way of decomposing permutations which we use in our proof. The domain of a permutation will be split into K equal size parts and the range intok such parts withk_≤K.

We start with some auxiliary notation. Recall that [a] denotes all in- tegers from 1 to a. We extend this notation by writing [a]_i/b for the set of

all integers k _∈ [a] such that i ₋1 < k/_ba/b_{c ≤} i, i.e., [a]i/b is the i-th

part after dividing [a] into b equal-sized parts (with b+ 1-st part containing the remaining elements). For example, [25]_2/6 = _{5,6,7,8_}. Observe that

|[a]1/b|=· · ·=|[a]b/b|=ba/bc and|[a]b+1/b| ≤b−1.

Fix now a permutation π of order n and integers K ∈ [n], i ∈ [K], k_∈[K] and j_∈[k]. We define Ri,j(π) as

Ri,j(π) ={x∈[n]i/K such thatπ(x)∈[n]j/k}

and we set

ρi,j(π) = |

Ri,j(π)| bn/K_c .

Vaguely speaking, ρi,j(π) ∈ [0,1] is the density of π in the part of the K×k

grid at the coordinates (i, j). The values of K and k will always be clear from the context.

To get used to the definition of the setsRi,j and the quantities ρi,j, we

now prove a simple auxiliary lemma.

Lemma 36. Let kandK be positive integers and letε0_≤1/(k+1)be a positive real. For every permutation π of order at least k(k+ 1)K and every x ∈[K], there existsy_∈[k] such thatρx,y(π)≥ε0.

Proof. Observe that

|Rx,1(π)|+· · ·+|Rx,k(π)| ≥ b|π|/Kc −k ≥ 1₋ 1 k+ 1 b|π_|/K_c.

Sinceε0 ≤1/(k+ 1), there must existy such thatρx,y(π)≥ε0 by the pigeonhole

principle.

Fix a permutationπ, integers k,K and M such that 1_≤k_≤K _{≤ |}π_|, and a real 0≤ε0 <1. If A is a k-sequence, then we say that a K-sequence B is (A, M, ε0)-approximateforπ if the following holds:

• the length ofB is|A|,

• B is monotone,

• |B_||B|=P |B|

i=1|Bi| ≥K−M, and

• for everyi∈[|A|], if x∈Bi and y∈[k]\Ai, thenρx,y(π)< ε0.

In other words, an (A, M, ε0)-approximateK-sequenceBdecomposes the whole index set [K] except for at mostM indices into|A|parts such that the indices

contained in the parts determined by B are in the increasing order and for x∈Bi, the only dense sets Rx,y(π) are those withy∈Ai.

Suppose that ak-sequenceA is_P-bad for a hereditary property_P. We say that aK-sequenceB is (A, ε0)-witnessingforπ if the following holds:

• the length ofB is_|A_|,

• there exist integers 1 _≤ x1 < . . . < x|A||A|·hAi ≤ K such that xj ∈ Bi if

|A_|i−1hAi< j≤ |A|ihAi, and • ρ_x

j,gA,hAi(j)(π)≥ε

0 _{for everyj}

∈[_|A_||A|·hAi] (the definition of the function

gcan be found in Section 6.1).

In other words, aK-sequenceB which decomposes the index set [K] is (A, ε0)- witnessing, if it is possible to find indices such that there are_|Ai|hAiindicesxj

in eachBiand all the setsRxj,gA,hAi(j)(π) are dense. The motivation for this def- inition is the following: if B is (A, ε0)-witnessing, then each set R_xj,gA,hAi_(j)(π) has at leastε0_{b|π|/Kcelements and consequently at least (ε}0_b|π|/Kc)|A|hAi

sub- sets of [_|π_|] induce subpermutations that are _hA_i-expansions of A. This will allow us to deduce that a random subpermutation of sufficiently large order does not have the property_P with high probability.

We now state a lemma saying that if aK-sequenceB is approximate but not witnessing with respect to ak-sequenceA for a permutation π, then there exists a reduction A0 _{of A and a K-sequence B}0 _{such that B}0 _{is approximate}

with respect toA0.

Lemma 37. Let _P be a hereditary property, let k, K, m and M be positive integers and letε0 ≤1/(k+1)be a positive real. LetAbe aP-badk-sequence and

B a monotone K-sequence with _|A_|= _|B_|. If the K-sequence B is (A, M, ε0_)-

approximate for a permutation π, _|π_{| ≥}k(k+ 1)K, B is not (A, ε0)-witnessing for π and _|Bi| ≥ mkhAi for every i ∈[|B|], then there exist a P-reduction A0 of A and a monotone K-sequence B0 _{such that}

• the lengths of A0 _{and B}0 _{are the same,}

• B0 _is(A0_{, M +mkhAi, ε}0_{)-approximate for π, and} • |B_i0| ≥m for every i∈[|B0|].

Proof. If B is not (A, ε0)-witnessing for π, then there exists an index j ∈

[_|B_|] such that there is no _|Aj|hAi-tuple x1 < · · · < x|Aj|hAi in Bj satisfying ρxi,yi(π)≥ε

0 _wherey

i=gA,hAi(|A|j−1hAi+i). Fix such an indexj for the rest of the proof.

If_|Aj|= 1, then anhAi-tuple with the properties given in the previous

paragraph is formed by any hAi elements of Bj by Lemma 36. So, we assume

that_|Aj| ≥2 in the rest of the proof. Definex1 to be the smallest index inBj

such that ρx1,y1(π) ≥ ε

0_{. Suppose that we have defined the indices x}

1, . . . , xi

and definexi+1 to be the smallest index in Bj that is larger than xi such that

ρxi+1,yi+1(π) ≥ ε

0_{. If no such index exists, we stop constructing the sequence.}

Let`be the number of the indices defined. By the choice ofj,` <_|Aj|hAi. For

completeness, setx0 = 0 and x`+1 =K+ 1.

DefineCi,i∈[`+ 1], to be the set of the elements ofBj strictly between

xi−1 and xi. If the subset Ci has size less than m, remove it from the sequence

and letC₁0, . . . , C_`00 be the resulting sequence. Observe that

|Bj| − P`0 i=1|C 0 i| ≤ `+ (`+ 1)(m−1) ≤ (`+ 1)m₋1 ≤ m_|Aj|hAi −1 ≤ mk_hA_{i −}1 (6.1)

since the setsC₁0, . . . , C_`00 contain all the elements ofB_j except for the elements

x1, . . . , x`and the elements contained in the setsC1, . . . , C`+1with cardinalities at mostm₋1. In particular, we can infer from_|Bj| ≥mkhAi that`0 ≥1.

Next, define C_i00, i_∈ [`0], to be the set of y <sub>∈</sub>[k] such that there exists x∈C<sub>i</sub>0 with ρx,y(π)≥ε0. Lemma 36 implies that the sets C100. . . , C`000 are non- empty. We infer from the way we have chosen the indicesx1, . . . , x` that each

set C_i00 is a proper subset of Aj. Finally, define the k-sequence A0 to be the

K-sequence A with Aj replaced with C100, . . . , C`000 and the K-sequence B0 to be the K-sequence B with Bj replaced with C10, . . . , C`00. By the definition of

C₁00, . . . , C_{`</sub>000 and by (6.1), theK-sequenceB0 is (A0, M+mkhAi, ε0)-approximate forπ. By the choice ofC<sub>1</sub>0, . . . , C<sub>`}00, we have that |B0_i| ≥m for every i∈[|B0|]. Finally, since`0<sub>≤`≤ |A</sub>

j|hAi and everyCi00 ,i∈[`0], is a proper subset ofAj,

A0 is_P-reduction of A.

We finish this section with the following lemma on approximating the structure of a sufficiently large permutation π with respect to a hereditary property.

Lemma 38. Suppose _P is a hereditary property. For all integers k and reals

ε and ε0 such that 0 < ε _≤ 1 and 0 < ε0 _≤ 1/(k+ 1), there exists an integer

K such that for every permutation π of order at least k(k+ 1)K, there exist a

• A is_P-bad and B is(A, ε0_{)-witnessing for π, or} • A is_P-good andB is (A,_bεK_c, ε0_{)-approximate forπ.}

Proof. Let _T be the k-branching with respect to_P. Let dbe the depth of _T, i.e., the maximum number of vertices on a path from the root to a leaf, and let w0 be the weight of the root of T. We show that K := ddw0/εe has the properties claimed in the statement of the lemma.

Let π be a permutation of order at least k(k+ 1)K. Based on π, we define a path from the root to one of the nodes in T in a recursive way. In addition to choosing the nodes ui _{on the path, we also define monotone K-}

sequencesBi _{such that B}i _{is (A}ui

, i_·w0, ε0)-approximate for π and |Bji| ≥ wui for everyj∈[|Bi_|].

Letu0 _{be the root ofT and setB}0 _{to be the basicK-sequence. Clearly,} B0 _{is (A}u0

,0, ε0)-approximate forπ. Suppose that the node ui _{on the path has}

already been chosen and we now want to choose the next node. If ui _{is a leaf}

node, we stop. Ifui _{is not a leaf node, then thek-sequenceA}ui

must be_P-bad. IfBi _{is (A}ui

, ε0)-witnessing for π, we also stop. Otherwise, Lemma 37 applied withmequal to the maximum weight of a child ofui_{(note that|B}i

j| ≥mkhAu

i

i

for everyj _∈[_|Bi_{|]) implies that there exist a P-reduction A}0 _{of A}ui

and a K- sequenceBi+1 _{such thatB}i+1_{is (A}0_{, i·w}

0+mkhAu i

i, ε0)-approximate forπ and

|Bi+1

j | ≥m for every j ∈[|Bi+1|]. Choose ui+1 to be the child of ui such that

Aui+1 =A0. Since mk_hAui i ≤w0, we obtain thatBi+1 is (Au i+1 ,(i+ 1)w0, ε0)- approximate forπ.

Let ` be the length of the constructed path. We claim that the k- sequence Au`

and the K-sequence B` _{have the properties described in the}

statement of the lemma.

Ifu` <sub>is not a leaf node, thenA</sub>u`

is_P-bad andB` <sub>is (A</sub>u`

, ε0_)-witnessing

forπ (since we have stopped at u`<sub>). Ifu</sub>` _{is a leaf node andA}u`

is_P-bad, then B` <sub>is (A</sub>u`

, ε0)-witnessing for π by Lemma 37 applied for m = 1 (Au`

cannot have a_P-reduction because it is simple). Finally, ifu` <sub>is a leaf node andA</sub>u`

is

P-good,B` <sub>is (A</sub>u`

,_bεK_c, ε0)-approximate for π sincedw0 ≤ bεKc.