unknown - elementary submodels in infinite combinatorics

7/27/2019 Unknown - Elementary Submodels in Infinite Combinatorics

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arXiv:1007

.4309v2

[math.LO

]6Dec2010

ELEMENTARY SUBMODELS IN INFINITECOMBINATORICS

LAJOS SOUKUP

Abstract. The usage of elementary submodels is a simple butpowerful method to prove theorems, or to simplify proofs in infi-nite combinatorics. First we introduce all the necessary conceptsof logic, then we prove classical theorems using elementary sub-models. We also present a new proof of Nash-Williamss theoremon cycle-decomposition of graphs, and finally we improve a decom-position theorem of Laviolette concerning bond-faithful decompo-sitions of graphs.

1. Introduction

The aim of this paper is to explain how to use elementary submodelsto prove new theorems or to simplify old proofs in infinite combina-torics. The paper mainly addresses novices learning this technique: weintroduce all the necessary concepts and give easy examples to illus-trate our method, but the paper also contains new proofs of theoremsof Nash-Williams on decomposition of infinite graphs, and an improve-ment of a decomposition theorem of Laviolette concerning bond-faithfuldecompositions.

The first known application of this method is due to Stephen G.Simpson, (see [16] and the proof of [3, Theorem 7.2.1]), who proved theErdos-Rado Theorem using this technique, and indicated that one cangive similar proofs for several other known theorems of combinatorialset theory ...

Our aim is to popularize a method instead of giving just black boxtheorems.

In section 2 we recall and summarize all necessary preliminaries fromset theory, combinatorics and logic.

Date: December 7, 2010.2000 Mathematics Subject Classification. 03E35, 54E25.Key words and phrases. graphs, infinite, elementary submodels, decomposi-

tion, cycle-decomposition, bond-faithful decomposition, Double Cover Conjecture,Fodors Lemma, Pressing Down Lemma, -system, partition theorem, free subset.

The author was supported by the Hungarian National Foundation for ScientificResearch grants no. K 68262 and K 61600.

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2 L. SOUKUP

In section 3 we give the first application of elementary submodels,and we explain why it is natural to consider -elementary submodels

for some large enough finite family of formulas.In section 4 we use elementary submodels to prove some classicaltheorems in combinatorial set theory. All these theorems have thefollowing Ramsey-like flavor: Every large enough structure containslarge enough nice substructures.

In section 5 we prove structure theorems of a different kind: Everylarge structure having certain properties can be partitioned into smallnice pieces. A typical example is Nash-Williamss theorem on cycledecomposition of graphs without odd cuts. To prove these structuretheorems it is not enough to consider just one elementary submodel butwe should introduce the concept of the chains of elementary submodels.

Finally, in section 6, we give a more elaborate application of chainof elementary submodels to eliminate GCH from a theorem concerningbond-faithful decomposition of graphs.

This paper addresses persons who are interested in infinite combi-natorics, but who are not set theory specialist. If you want to studymore elaborated applications of these methods, see the survey papersof Dow [5] and Geshcke [6], or the book of Just and Weese [10, Chapter24]. These papers are highly more technical, than the current one, butthey also contain many applications in set theoretic topology.

For applications of these methods in infinite combinatorics, see also

[2], [7], [8] and [11]. Chains of elementary submodels play also a crucialrole in the proof of some key results of the celebrated pcf theory ofShelah, see [15] or [1].

2. Preliminaries

2.1. Set theory. We use the standard notions and notation of settheory, see [9] or [12]. If is a cardinal and A is a set, let

(1)

A


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ELEMENTARY SUBMODELS 3

Fact 2.1. V = {V : On}, i.e. for each set x there is an ordinal such that x V.

2.2. Combinatorics. We use the standard notions and notation ofcombinatorics, see e.g. [4]. A graph G is a pair V(G), E(G), where

E(G)

V(G)2

. V(G) and E(G) are the sets of vertices and edges,respectively, of G. We always assume that V(G) E(G) = .

A -cover of a graph G is a family G of subgraphs of G such thatevery edge of G belongs to exactly members of the family G. Adecomposition is a 1-cover, i.e. a family G such that {E(G) : G G}is a partition of E(G).

If M is a set then let

G[M] =

V(G) M, E(G) M2

; G M = V(G), E(G) \ M .

So GM denotes the graph obtained from G removing all edges in M.If x, y(x, y M {x, y} M), then the graphs G[M] and G Mform a decomposition of G.

If G is fixed, and A V(G) then we write A for V(G) \ A. A cut ofG is a set of edges of the form E(G) [A, A] for some A V(G). Abond is a non-empty cut which is minimal among the cuts with respectto inclusion.

Fact 2.2. = F E(G) is a bond in G iff there are two distinctconnected components C1 and C2 of G F such that F = E(G) [C1, C2].

The following statement will be used later several times.

Proposition 2.3. Assume that H is a subgraph of G, F is a bond

in H. If F is not a bond in G then F

D2

for some connectedcomponent D of G F.

Proof. By Fact 2.2 there are two distinct connected components C1 andC2 of H F such that F = E(H) [C1, C2]. IfC1 and C2 are subsetsof different connected components of G, C1 D1 and C2 D2, then

F = [C1, C2]E(H) [D1, D2]E(G) F([D1, D2]E(G\F)) = F,

i.e. F = [D1, D2] E(G) and so F is a bond in G by Fact 2.2 above,which contradicts the assumptions. So C1 and C2 are subsets of the

same connected component D ofGF. Thus F [C1, C2]

D2

.

Given a graph G for x = y V(G) denote by G(x, y) the edgeconnectivity of x and y in G, i.e.

G(x, y) = min{|F| : F E(G) : F separates x and y in G}.


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4 L. SOUKUP

By the weak Erdos-Menger Theorem there are G(x, y) many edge dis-joint paths between x and y in G.

2.3. Logic. The language of set theory is the first order language Lcontaining only one binary relation symbol . So the formulas ofL areover the alphabet {, , (, ), =, } Var, where Var is an infinite setof variables. To simplify our formulas we often use abbreviations likex, , x y, !x, x y , etc.

An L-structure is a pair M, E, where E M M. In this paperwe will consider only structures in the form M, M where M isthe restriction of the usual membership relation to M, i.e.

M = {x, y M M : x y}.

We usually write M, or simply M for M, M.If(x1, . . . , xn) is a formula, a1, . . . , an are sets, then let (a1, . . . , an)

be the formula obtained from (x1, . . . , xn) by replacing each free oc-currence of xi with ai. [An occurrence of xi is free it is not within thescope of a quantifier xi.]

If (x, x1, . . . , xn) is a formula, a1, . . . , an are sets, then C = {a :(a, a1, . . . , an)} is a class. Especially, every set b is a class: b = {a :a b}. Moreover, all sets form the class V: V = {a : a = a}. In thispaper we will consider just these classes: the sets and the universalclass V.

For a formula (x1, . . . , xn), a class M, and for a1, . . . , an M wedefine when

(2) M |= (a1, . . . , an),

i.e. when M satisfies (a1, . . . , an), by induction on the complexity ofthe formulas in the usual way:

(i) M |= ai aj iff ai aj ,(ii) M |= iff M |= or M |= .

(iii) M |= iff M |= fails.(iv) M |= x(x, a1, . . . an) iff there is an a M such that M |=

(a, a1, . . . , an)

For a formula (x1, . . . , xn) let M(x1, . . . , xn) be the formula obtained

by replacing each quantifier x with x M in . Clearly for eacha1, . . . , an M,

(3) M(a1, . . . , an) iff M |= (a1, . . . , an).

If(x1, . . . , xn) is a formula, M and N are classes, M N, then wesay that is absolute betweenM and N,

(4) M N


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in short, iff for each a1, . . . , an M

(5) M |= (a1, . . . , an) iff N |= (a1, . . . , an)

If is a collection of formulas then write

(6) M N

iff M N for each .M is an elementary submodel of N,

(7) M N

iff M N for each formula .If is absolute between M and V, then we say that is absolute

for M.

Theorem 2.4 (Lowenheim-Skolem). For each setN and infinite subsetA N there is a set M such that A M N and |M| = |A|.

Since ZF C Con(ZF C) by Godels Second Incompleteness Theo-rem, it is not provable in ZFC that there is a set M with M |= ZF C.So, since V |= ZF C, it is not provable in ZFC that there is a set Mwith M V. Thus, in the Lowenheim-Skolem theorem above, theassumption that N is a set was essential. However, as we will see, thefollowing result can serve as a substitute for the Lowenheim-Skolemtheorem for classes in certain cases.

Theorem 2.5 (Reflection Principle). Let be a finite collection offormulas. Then for each cardinal there is a cardinal such thatV V, and

V is regular then there is a set M V with x M,

|M| < and M .(3) If = then there is a set M V such that x M, |M| = ,M + +, and

M

M.(4) If > is regular then the set

Sx = {M : x M V, M }

contains a closed unbounded subset of .


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6 L. SOUKUP

Proof. Fix a cardinal with x V. By the Reflection Principlethere is a cardinal > such that V V and

V

V.

(1) Straightforward from the Lowenheim-Skolem theorem: since V isa set, |V| , and x V there is M V with x M and |M| = .Then M V.

(2) Construct a sequence Mn : n < of elementary submodels of Vwith |Mn| < as follows. Let M0 be a countable elementary submodelof V with x M. If Mn is constructed, let n = sup(Mn ). Since is regular we have n < . By the Lowenheim-Skolem theorem thereis an elementary submodel Mn+1 ofV such that Mn n Mn+1 and|Mn+1| = |Mn n| < . Finally let M = {Mn : n < }. ThenM V, and so M V, and M = sup n .

(3) Construct an increasing sequence M : < 1 of elementary sub-models of V with |M| = as follows. Let M0 be an elementarysubmodel of V with {x} M0 and |M0| = . For limit letM = {M : < }. If M is constructed, let = sup(M

+).Since |M| = we have <

+. Let X = M

M

. Then|X|

= . By the Lowenheim-Skolem theorem there is an elemen-tary submodel M+1 of V with X M+1 and |M+1| = . Finallylet M = {M : < 1}. Since 1, M

+ = sup{ : |M|, we can pick A A \ M. Let D = M A.Then D

M

. Since

M


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Let C = B {A}. If B B, then B M and so B M andA B = D by M V. So C B is a -system with kernel D.

Contradiction. Next we prove two classical partition theorems. First we recall (a

special case of) the arrow notation notation of Erdos and Rado. As-sume that , and ordinals. We write

(25) (, )2

iff given any function f :

2

2 either there is a subset B of

order type with f

B2

= {0}, or there is a subset C of order

type with f

C2

= {1}.

Theorem 4.4 (ErdosDusnikMiller). If = cf() > then

(, + 1)2.

Proof. Fix a coloring f :

2

2.Let be a large enough finite set of formulas. By Corollary 2.6(2)

there is a set M with |M| < such that f M V and M .

01 A

BC

M

Figure 1.

Fix \ M. Let A be a -maximal subset of M such thatA {} is 1-homogeneous. If A is infinite, then we are done.

Assume that A is finite. Let

(26) B = { \ A : A f(, ) = 1}.

Clearly B. Since f, A M we have B M. Let C B be a-maximal 0-homogeneous subset.Claim: |C| = .Assume on the contrary that |C| < . Then |C| M and so

|C| M because M . Thus C M by Claim 3.7. Let C.Since M\ A we have that A {} {} is not 1-homogeneous. ButA {} is 1-homogeneous and B, so f(, ) = 0. Thus C {} is0-homogeneous. Since B, we have C by the maximality of C,which contradicts B M.

Theorem 4.5 (ErdosRado). c+ (c+, 1 + 1)2.


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14 L. SOUKUP

Proof. Fix a function f :c+2

2.Let be a large enough finite set of formulas. By Corollary 2.6(3)

there is a set M with |M| = c such that f M V, M c+ c+and

M

M.Pick c+ \ M.Let A be a -maximal subset of M such that A {} is 1-

homogeneous. If A is uncountable, then we are done.Assume that A is countable. Since

M

M, we have A M.Let

(27) B = { \ A : A f(, ) = 1}.

Since f, A M we have B M. Let C B be a -maximal 0-homogeneous subset.

Claim: |C| = c+.Assume on the contrary that |C| c. Then |C| c M and soC M by Claim 3.7. Let C. Since M \ A we have thatA {} {} is not 1-homogeneous. But A {} is 1-homogeneousand B, so f(, ) = 0. Thus C {} is 0-homogeneous. Since B, we have C by the maximality of C, which contradictsB M.

Given a set-mappingF : X P(X) we say that a subset Y X isF-free iff y / F(y) for y = y Y.

Theorem 4.6. If = cf() > and F :

, f : is a regressive functionthen there is < such that f1{} is unbounded in .


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Proof. Let be a large enough finite set of formulas. By Corollary2.6(2) there is a set M with |M| < such that f M V and

M .Let = sup(M ) and consider = f(). We claim that T =f1{} is unbounded in . Since = M we have T N. If Tis bounded, then sup T M = . However T, so T should beunbounded.

Theorem 4.8. (Fodors Pressing Down Lemma) If = cf() > ,S is stationary, and f : S is a regressive function then thereis an ordinal < such that f1{} is stationary.

Proof. Let be a large enough finite set of formulas. By Corollary2.6(4) there is a set M with |M| < such that S, f M V and

= M S.Let = f(). We show that T = f1{} is stationary. Clearly

T M. If T is not stationary then there is a closed unbounded setC M such that C T = .

Claim: sup(M ) C if C M is a closed unbounded subset of .Since C is closed, if sup(M ) / C then there is < sup(M ) suchthat (C\ ) M = . Then M |= C\ = . Thus V |= C\ = ,i.e. C , which contradicts the assumption that C is unbounded.

So by the claim C T. Contradiction.

5. Decomposition theorems

In the previous section we proved theorems which claimed that Givena large enough structure A we can find a large enough nice substructureof A. In this section we prove results which have a different flavor:Every large structure having certain properties can be partitioned intonice small pieces.

In [14] the following statements were proved:

Theorem 5.1 (Nash-Williams). G is decomposable into cycles if andonly if it has no odd cut.

We give a new proof which illustrates how one can use chains ofelementary submodels. To do so we need two lemmas. The first onewas proved in section 3:

Lemma 3.5. If G = W, E is an NW-graph, G M V for somelarge enough finite set of formulas, then G[M] is also an NW-graph.

The second one is the following statement.


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16 L. SOUKUP

Lemma 5.2. If G = W, E is an NW-graph, G M V for somelarge enough finite set of formulas, thenGM is also an NW-graph.

Lemma 5.2 above follows easily from the next one.

Lemma 5.3. Assume that M V with |M| M for some largeenough finite set of formulas. If G M is a graph, x = y V(G)and F E(G M), such that

(28) |F| |M|, GM(x, y) > 0 and F separates x and y in G M

then

(29) F separates x and y in G.

Proof of Lemma 5.2 from Lemma 5.3. Assume on the contrary that G

M has an odd cut F. Since any cut is the disjoint union of bonds wecan assume that F is a bond.

Pick c1c2 F. Then clearly GM(c1, c2) > 0. Moreover F separatesc1 and c2 in G M, so F separates them in G by Lemma 5.3, i.e. c1and c2 are in different connected components of G F

However F can not be a bond in G, so by Proposition 2.3 there is

a connected component D of G F such that F

D2

. i.e. c1 andc2 are in the same connected component of G F. This contradictionproves the lemma.

Proof of Lemma 5.3. Assume that G, M, x, y and F form a counterex-

ample.

M

F

M

x y

Q

Q

Qx Qy

P

PR

R

G

qjx qjy

Figure 2.

Fix a path P = p0p1 . . . pn from x to y in G M which witnessesthat GM(x, y) > 0, i.e. p0 = x, pn = y and pipi+1 E(G) \ M fori < n.


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We assumed that F does not separate x and y in G, so there is apath Q = q0 . . . q m from x to y witnessing this fact, i.e. q0 = x, qm = y

and qjqj+1 E(G) \ F for j < m. Since F separates x and y in GMthere is at least one j < m such that qjqj+1 M.Let jx = min{j : qj M} and jy = max{j : gj M}. Since jx j

and jy j + 1 we have jx < jy. Let x = qjx and y = qjy . Let

Qx = qjxqjx1 . . . q 1q0 and Qy = qmqm1 . . . q jy . Then QxP Qy is a walkfrom x to y in G M. Hence GM(x

, y) > 0.

Claim: G(x, y) > |M|.

Indeed, assume that = G(x, y) |M|. Since M V and

x, y M there is A M

V(G)

such that A separates x and y

in G. Since |A| = M we have A M. So M separates x and y,i.e. GM(x

, y) = 0. This contradiction proves the claim.

By the weak Erdos-Menger Theorem there are G(x, y) many edge

disjoint paths between x and y in G. Since |MF| = |M| < G(x, y)there is a path R = r0 . . . rk from x

to y which avoids M F. ThenQ1x RQ

1y is walk from x to y in GM which avoids F. Contradiction.

Proof of theorem 5.1. We prove the theorem by induction on |V(G)|.IfG is countably infinite then for each e E(G) there is a cycle C in

G with e E(C) because e is not a cut in G. Moreover, G C is alsoan NW-graph, i.e. it does not have odd cuts. Using this observation

we can construct a sequence {Ci : i < } of edge disjoint cycles in Gwith E(G) = {E(Ci) : i < }.Assume now that = V(G) > and we have proved the statement

for graphs of cardinality < .Let be a large enough finite set of formulas. By the Reflection

Principle 2.5 there is a cardinal such that V V and

V

V.Then G V.

We will construct a sequence M : < V of elementary sub-models of V with

() |M| = + ||, M and M M+1

as follows:(i) Let M0 be a countable elementary submodel of V with G M0.

(ii) if < is a limit then let M = {M : < }. Since |M| + || < and M V we have M V.

(iii) If = + 1 then |M {M} | = + || so by Lowenhein-Skolem Theorem there is M V with M {M} Mand |M| = + ||.


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18 L. SOUKUP

The construction clearly guarantees (). Using the chain M : < decompose G as follows:

for < let G = (G M)[M+1].By Lemma 5.2 the graph G = G M is NW. Moreover, since M M+1 we have GM M+1. So we can apply Lemma 3.5 for M+1and G to deduce that G is NW.

So we have decomposed the graph G into NW-graphs {G : |M| then dG(x) = dGM(x). SodGM(x) can not be an odd natural number.

(2) Assume that in G there is no finite component. Let G M Vwith |M| M for some large enough finite set of formulas.Claim There is no finite non-trivial component in G[M].

Let x V(G) M and assume that x has a finite component C inG[M]. Then C M and

(34) M |= C is the component of x,

so

(35) V |= C is the component of x,

i.e. G has finite component.Claim There is no finite non-empty component in G M.

Assume that there is a finite non-trivial component C in G M.Since C is not a component in M there is an edge cd E(G) M withc C. Since C is non-trivial there is c C such that cc is an edge inG M. Then c M and c / M.

Since dG(c) |M| would imply c {c : cc E(G)} M we havedG(x) > |M|. However {c : cc E(G)} \ M C, and so |C| > |M|.Contradiction.

We want to apply Theorem 5.6. Let i be the property NWi fori = 1, 2, and 1 be decomposable into cycles and endless chains ,and 2 be decomposable into endless chains .


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22 L. SOUKUP

Then condition 5.6.(1) holds by Lemma 5.13, 5.6.(2) is true byLemma 5.12. 5.6.(3) is trivial from the definition. Putting these things

together we obtain the theorem.

6. Bond faithful decompositions

In this section we prove a decomposition theorem in which we cannot apply Theorem 5.6.

Definition 6.1. Let be an infinite cardinal. A decomposition H ofa graph G is -bond faithful iff |E(H)| for each H H,

(i) any bond ofG of cardinality is contained in some member ofthe decomposition,

(ii) any bond of cardinality < of a member of the decomposition is

a bond ofG.Theorem 6.2 (Laviolette, [13, Theorem 3]). Every graph has a bond-

faithful-decomposition, and with the assumption of GCH, every graphhas a bond-faithful -decomposition for any infinite cardinal .

Applying methods of elementary submodels leads more naturally toa simpler proof of the theorem above that does not rely on GCH.

Theorem 6.3. For any cardinal every graph has a -bond faithfuldecomposition.

The following lemma is the key to the proof.

Lemma 6.4. Let G be a graph, G M V with = |M| M forsome large enough finite set of formulas.

(I) If F E(G[M]) is a bond of G[M] with |F| < |M| then F is abond in G.

(II) If F E(G) is a bond of G M with |F| < |M| then F is abond in G.

Proof of 6.4. (I) Assume on the contrary that F is not a bond in G.Pick xx F. Then by Proposition 2.3 x and x are in the sameconnected component D ofGF, and so there is a path P = x1x2 . . . xn,in G F, x1 = x, xn = x

. Choose the path in such a way that the

cardinality of the finite set(36) IP = {i : xixi+1 / M}

is minimal. Since F is a cut in G[M] we have IP = . Let i = min Ip.Then xi M. Let j = min{j > i : xj M}. Then j > i + 1,xi, xj M, and moreover (GM)F(xi, xj ) > 0.

Claim 6.5. If x, y M, GM(x, y) > 0 then G[M](x, y) = |M|.


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Proof of the Claim. There is a vertex set A

V(G)G(x,y) such that

A separates x and y in G. We assumed that is large enough, espe-

cially it contains the formula A (A,x,y,G), where (A,x,y,G) isthe following formula:

A

V(G)G(x,y) is a vertex set which separates x and y in G.

Since M V, and the parameters G,x,y are in M, there is an Ain M such that M |= (A,x,y,G). Since we assumed that is largeenough, it contains the formula (A,x,y,G). So V |= (A,x,y,G),

i.e. A

V(G)G(x,y) M is a vertex set which separates x and y in

G.If G(x, y) M then A M implies A M by Claim 3.7, and

so M separates x and y in G. Thus GM(x, y) = 0.But GM(x, y) > 0, so we have G(x, y) > |M|. So, by the weakErdos-Menger Theorem there is a family P of many edge disjointpaths between x and y in G. Since G,x,y, M we can find such aP in M. But |P| = M, and so P M. Thus there are -manyedge disjoint paths between x and y in M, i.e. G[M](x, y) = .

By the Claim G[M](xi, xj) = . So, by the weak infinite MengerTheorem, there are many edge disjoint path in G[M] between xi andxj . Since |F| < , there is a path Q = xiy1 . . . ykxj in G[M] whichavoid F. Then P = x1 . . . xj y1 . . . ykxj . . . xn is a path between x1 andxn in G F with |IP| < |IP|. Contradiction.(II) Let c1c2 F. Then GM(c1, c2) > 0, F separates c1 and c2 inG M, so F also separates c1 and c2 in G by Lemma 5.3. In otherwords, c1 and c2 are in different connected component of G F, andso F should be a bond in G by Proposition 2.3.

Proof of Theorem 6.3. By induction on |V(G)|. If|V(G)| then theone element decomposition {G} works.

Assume that G = , E, and > . Let be a large enough finiteset of formulas. By the Reflection Principle 2.5 there is a cardinal such that V V and

V

V.By Fact 5.5 there is a -chain of elementary submodels M : <

of V with G, M0. Since and < and M for < , we canassume that M0.

Using the chain M : < partition G as follows:

for < let G = (G M)[M+1].

Let G = G M. By Lemma 6.4(II)

any bond of cardinality < of G is a bond of G.


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24 L. SOUKUP

Moreover, since M M+1 we have GM M+1. So we can applyLemma 6.4(I) for M+1 and G

to derive that

any bond of cardinality < of G is a bond of G.Putting these together

any bond of cardinality < of G is a bond of G.

Moreover |V(G)| |M+1| + || < , so by the inductivehypothesis, every G has a -bond faithful decomposition H. LetH = {H : < }. H clearly satisfies 6.1(ii): if F is a bond of someH H with |F| < , then F is a bond ofG, and so F is a bond ofG by ().

Finally we show that H satisfies 6.1(i) as well. We recall one moreresult of Laviolette:

Theorem 6.6 ([13, Proposition 3]). For any cardinal every graphhas a decomposition K which satisfies 6.1(i) and |E(K)| for eachK K.

Let us remark that GCH was assumed in [13, Proposition 3], but inthe proof it was not used.

Let (G, , K) be the following formula:

K is a decomposition of G which satisfies 6.1(i)

and |E(K)| for each K K.

Since was large enough we can assume that it contains the formulas(G, , K) and K(G, , K). Since M0 V, and G, M0 wehave a K M0 such that (G,, K) holds, i.e. K is a decompositionof G which witnesses 6.1(i) and |E(K)| for each K K. Assumethat A is a bond of G with |A| . Then there is K K suchthat A E(K). Let be minimal such that E(K) M+1 = ,and pick e E(K) M+1. Then K is definable from the parametersK, e M+1 by the formula K K e K. So K M+1 by Claim2.7. Thus A E(K) E(G). Since, by the inductive assumption,the decomposition H satisfies 6.1(i) there is H H with A E(H).But H H, so we are done.

References

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Alfred Renyi Institute of Mathematics, Hungarian Academy of Sci-

ences, Budapest, Hungary

E-mail address: [email protected]: http://www.renyi.hu/soukup
http://arxiv.org/abs/1004.0181http://arxiv.org/abs/1004.0181

unknown - elementary submodels in infinite combinatorics

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