Minimal types in simple theories(2)

We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor-Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T

M0,M0|=φ(a)}.ThesetXde nedinM0bysuchaformulaφ(x)iswhatissometimescalledminimal:namelyitisin nite,buteveryde nable(withparametersinM0)subsetis niteorco nite.IfM0isanin nitestructuresuchthatS(M0)iscountable,thentherewillexist1-typesoverM0ofCB-rank1.In[1]itwasobservedthatifTh(M0)isstableandp(x)∈S(M0)hasCB-rank1thenpisaregulartype:forkingonrealizationsofpinabig¯isapregeometry.HenceifMnisprimeoverM0togetherwithanmodelM

independentset{a1,..,an}ofrealizationsofp,then(bytheopenmappingtheoremandregularity),n=mimplesMnandMmarenotisomorphic(overM0).TheresultsinthispapershowthatthesameconstructionworksifTh(M0)isjustassumedtobesimple.

Ontheotherhand,themaintechnicalresultofthispaper(stationarityofCB-rank1typesovermodelsinsimpletheories)suggeststhatifTisacountablesimpletheorysuchthatforsomecountablemodelM0ofT,S(M0)iscountable,thenTisclosetobeingstable.OfcourseifthemodelM0happenstobeω-saturated(soforexamplewhenTisω-categorical),countabilityofS(M0)directlyyieldsω-stabilityofT.Buthereisanunstableexample(alsosuggestedbyB.Kim):Eisanequivalencerelation,andinM0thereisexactlyoneequivalenceclassCnofeach nitecardinalityn(andnoin niteclasses).AlsoRisabinaryrelation,andthe nitestructures(Cn,R|Cn)approximatetherandomgraph,sothatifCisanin niteE-classinanelementaryextensionMofM0then(C,R|C)istherandomgraph.ThiscanbesetupsothatS(M0)iscountable(andinfactsuchthatthereisauniquenon-algebraic1-typeoverM0).

2Proofofmainresult

AssumeTtobeacompletesimpletheoryinalanguageLofarbitrarycar-dinality.Weassumefamiliaritywiththebasicmachineryofstabilityandsimplicity(see[2]and[5]).Wewillmakeheavyuse(amongotherthings)ofthefactthatpisatypeoveramodelthenanyheirorcoheirofpisanonforkingextension.¯ofT.AllothermodelsofTweWeworkinabigsaturatedmodelM

considerareassumedtobesmallelementarysubmodelsofT.

Proposition2.1LetM0beamodelofT.Letp(x)∈S(M0)beatypeof

2