RAMSEY CLASSES SPARSITY AND MODELS FOR FINITE - NESIETPIIL - PDF document

RAMSEY CLASSES SPARSITY AND MODELS FOR FINITE - NESIETPIIL JAROSLAV UNIVERSITY CHARLES PRAGUE WITH MENDEZ DE PATRICE OSSONA , HUBIEKA JAN EVANS DAVID , _ JAN 31,2018 PARIS IHP ,

1. CONTENTS SPARSITY & STABILITY 1. . FOR RAMSEY CLASSES 2 MODELS . . UNIVERSALITY 3. .

2 Dot GRAPHS E FINITE CLASS OF A IS SOMtWHerLDLN5E_T IF EVERY GRAPH EN FOR d SOME A IS d- MINOR SHALLOW OF E GRAPH IN . FHEE ( H >a6 ) tfG . - H OBTAINED BE FROM A CAN GRAPH G OF BY CONTRACTING SUB GRAPHS WITH RADIUS { d SOME SUB . = (G) Ed d } y ; distfxiy Nod kI={ ) < VC G) c- Ndo G) RADIUS ⇐7 XEV (G) FOR SOME

3 EQUIVALENTLY . : ± d=3 •I9• xQ€Q¥Q⇒ _ t#E→¥#¥ .IM#P.hEeeCY - Auorapt 's ) .

4 . DEFD Is NOWHEREDENT-lf.IT is C DENSE NOT SOMEWHERE . # EXPLICITELY : A GRAPH Gd DEN FOR IS EVERY THERE A BE SUCH FAILS TO Gd THAT OF d DEPTH AT SHALLOW MINOR @ IN A GRAPH . - E GRAPHS td g- Td ALL : JN P DE & OSSONA MENDEZ , . SPRINGER 2012 SPARSITY . ,

Ga EXAMPLES TREES � 1 � GRAPHS PLANAR � 2 � CLOSED PROPER MINOR � 3 � CLASSES EEEY.FFFE.to# } LY G) Ed { G ; � 4 � PLANAR GRAPHS G- QUASI � 5 � ERDOS CLASSES � 6 � e. g . ... } { G ,Gn . ,Gz , , ... PROPERTIES Gi WITH GRTHCG ;) . ) A ( Gn < i< . ) .X( G is -

5 . ? WHY SPARSITY " LINEARLY MANY " ALMOST EDGES : 6711+04 ) )l<_w( IECG . God Echl D (G) NCHI max = He d ) OF ( DENSITY EDGE MAXIMAL .my#E=O AT OF G MINOR SHALLOW A d DEPTH 'THM] ( JN 2008 ) POM / + @ FOLLOWING THE CLASS A FOR EQUIVALENT ARE : C DENSE @ NOWHERE IS d EVERY FOR � 2 � him . IVC G) I log Cec

6. SAME CLASSIFICATION FOR MINORS TOPOLOGICAL : G GRAPH A MINOR OF ' SUBDIVISION H A IS THERE EDGE EVERY WHERE H OF ATMOST BY SUBDIVIDED IS H OF ' 15 ASUBGRAPH H AND VERTICES ATDEPTHd.lt 2d G OF .hu/SHA-kOWt0P0L0G1CAL-_ . To EFD HIAG . , 2008 ) if J .N .tl?0.M THMLT . . IFF DENSE C NOWHERE Is EGD §FmYpµs d ANY FOR _ NOTIONS ROBUST DICHOTOMY DENSE SPARSE -

Ga 72 CHARACTERISATIONS OF DENSE NOWHERE × DENSE SOMEWHERE DICHOTOMY a- sgnfpg Tak ) 's a) ( d EVERY FOR 111 (EXPANS10Nhf-B0UNDtD C CLASS DEGENERATED Td A GRAPHS OF 111 Efcd ) M¥ . . . . •

6b €4 ( CHARACTERISATIONS DENSE NOWHERE OF EXPANSION 't BOUNDED ANB E CLASS FOR A ND QQ @ 's Gell legal limsut .=o fd logical � 2 � 61=0 eogEd( limsup - ttd loglcol � 3 � geq w( G) < a limsup td � 4 � GECOD CHARACTERISATION � 5 � & ) QUASIWIDE ( LOWTREEDEPTH X � 6 � DECOMPOSITION DENSITY VC � 7 � COMPLEXITY NEIBORHOOD � 8 � CHECKING MODEL � 9 � COUNTING @ CATEGORIES @ - DVORA GROHE KRA 'L ) 'K THOMAS , , HLINENY , GAJARSKY , KREUTZER , , REIDL TORUNCZYK ) PILIPCZUK , , GAGO ROSSMANITH DEMAINE , , , WOOD YANG D. ZHU , KIERSTED , , , , NORIN ROSSMAN ATSERIAS DAWAR ) , d d @

6C EXPANSION BOUNDED to BOUNDED , Xp BOUNDED X , ALGORITHMS LINEAR VS DENSE NOWHERE ALMOST W , Wp BOUNDED Tp , LINEAR ALGORITHMS ALMOST LINEAR n1+O( 1) - BE Ee ND - X UNBOUNDED W BOUNDED " ERDO "s CLASSES ' '

7. CONNECTION MODEL TO STABILITY THEORY - C GRAPHS FINITE A CLASS OF IS STABLED EVERY FOR IF ) 4( Eiy FORMULA N( 4,4 ) WHICH EXISTS THERE GRAPHS ALL HALF BOUNDS ALL REPRESENTED BY IN y GE @ GRAPHS . ai ,5n 5. TUPLES ,En ... , , , ... . G REPRESENT GRAPH HALF IN Gt4(ai,5j ) j is IF IFF .

8 . ) PODEWSKI ( 1978 ZIEGLER , ( ) , 1. 2014 ADLER ADLER H . THEN DENSE NOWHERE IS STABLE THMWQIFY 15 IT . € CLOSED MONOTONE E IS IF � 2 � ) AND SUBGRAPHS STABLE ON NOWHERE DENSE THEN IT IS . PROP STABLE FLAT SUPER ⇒ DENSE I NOWHERE On DENSE SOMEWHERE MONOTONE + STABLE + � 2 � I REPRESENTING , Y ) FORMULA y( × ( FINITE ANY � 1 � Me HALFIGRAPH CORD STABLE DENSE = NOWHERE GRAPHS CLASSES MONOTONE OF FOR .

9. FINITE REFINEMENT , TORUIJCZYK SIEBERTZ PILIPCZUK , ) ( 2017 THMLT f :N3→N FUNCTIONS ARE THERE :N→N g PROPERTIES : FOLLOWING THE WITH ,G Kthgq GRAPH WITH G IF A is - 4 ( tif ) FORMULA 15 WITH A IF • VARIABLES d FREE AND WITH QRANK of G tf ( Iiy ) IN REPRESENTS THEN ftp.dit ) E WITH GRAPHS HALF ONLY VERTICES . Ms * 72 ) PROOF FOF SEVERAL USES OF ND CHARACTERISATIONS COMPACTNESS INSTEAD OF LOCALITY GAIFMAN USES LEMMA . ÷ PRESENTLY PROOF : IN ADDED

10 . WORK FUTURE SETTING THEORETIC MODEL - CHARACTERISATIONS OTHER OF DICHOTOMY ) ( OF DENSE SPARSE - HEREDITARY MONOTONE nd - ) EMBEDINGS ( INTERPRETATIONS OF - CLASSES EXPANSION BOUNDED ? ) ( 18 LICS CHARACTERISED ( ) DIDEROT FRIDAY ON

11 . THEORY RAMSEY ITS IN RELEVANCE THEORETIC MODEL

12 . mmmm ] [ RAMSEY COMBINATORKS name #h molfEL.lt#oRYw # / fmmz TOPOLOGICAL ' Dynamical " " STRUCTURAL RAMSEY THEORY

13 . DEFD - STRUCTURES L tft CLASS OF A . = OBJECTS SUB WITH . ( Yf ) BEK ALL A , FOR SUB B OBJECTS OF . A ISOMORPHIC TO . IF K Is EXISTS THERE BEK EVERY FOR LAf-RAMSEY ERDOI.DE#NARI THAT SUCH EK ( ( B) I C- > . ( (A) = An U Gz PARTITION EVERY FOR B' E ( CB ) ioe{ 1,2 } AND THERE EXISTS SUCH ( Bf ) THAT Eaio . ishmael IS IT K if AEK EVERY RAMSEY FOR A . -

14 . K PASCAL LEEB THEORIE . . , V. RODL J . N DEUBER W . _ SUB OBJECTS EMBEDDINGS = - RAMSEY OF LINE THE OF TOP _ EXAMPLES CLASSICAL ( 6 ) ORDERS LINEAR |_RoPerte] - E SETS FINITE + - knineN}+suB { K = - GRAPHS " VW PARAMETER BUT NOT - " SETS JEWETT THM HALES - . BUT THM RADO NOT - SUITABLE A YES FOR SETD AXIOMAMZAMON [ ( mipic ) -

15 . BASIC BLOC BUILDING HARRINGTON ) THMD RELATIONAL FOR ANY LANGUAGE L RAMSEY CLASS A IS MONOTONE RESPECT TO WITH ftp.pgp.ytruo?0Ns tianya.fr#inrenswe EMBEDDING S . Mrs. -> # y7 • - -

16 . FINITE MODELS RAMSEY FOR . HUBIEKA TIF ( J .J.N 2016 ) . LINEARLY ALL THE CLASS OF ( RELATIONS CONTAINING L WITH ) SYMBOLS FUNCTION AND CLASS RAMSEY A IS . - AA =( A ,(RµiREL)•(f*ifeL),< REL ) , ( ftp.ifeDKF } , ( Rpg ; ( B Be = 0RDEREDL-STRUCTURES##t M$ -7 MB EMBEDDING A -713 F AN : is SATISFIES : IF IT INJECTIVE - ) -< NB Epa W.R.tn MONOTONE - Raf PRESERVES ALL - F( fact , xp ) )== . , . . - ... ,Fkp7 ) fµ( Fkn ) , " " PRESERVE CLOSURES EMBEDDING 5

17 . :@ ORDERED CLASS OF THE STEINER SYSTEMS IS RAMSEY . PR0OF(OUTLlNE@) SYSTEM ( X , B) STEINER : C ¥ ) - ! BEBFYYEB ) F txtyex _ ( f) ( px ) f → DEFINE : f- ( xiy )=B×y - USE ( ) REFINEMENT OF RAMSEY MODELS OF CLASSES + COMPLETION AND EXISTENCE SYSTEMS STEINER OF . D

18 . CHARACTERISATION TOWARDS CLASSES RAMSEY OF - AND / OR SYMMETRIES HIDDEN CLASSES RAMSEY OF non RAMSEY CLASS PARADOX : IMPLIES AND NEEDS RIGIDITY OTHER THE BUT ON CLASSES RAMSEY SIDE HIGHLY FROM COME SITUATION SYMMETRIC .

qq.GE?EmtmTm apass AMALGAMATION TL - CLASS RAMSEY f FRAISSE Thi ' - LIMIT

20 . HEREDITARY RAMSEY EVERY - EMBEDDING JOINT WITH CLASS AMALGAMATION PROPERTY AN IS CLASS . FRAISSE HOMOGENEOUS LIMIT 15 ULTRA - HOMOGENEOUS ULTRA CHARACTERISATION OF a RAMSEY OF CHARACTERISATION CLASSES ( TRUE ALL CASES IN WITH CHARACTERISATION KNOWN HOMOGENEOUS ULTHZA OF ) STRUCTURES ORDERS PARTIAL GRAPHS , , TOURNAMENTS oo . , PROGRESS WORK IN _ HOMOGENEOUS : ULTRA FOR CHERLIN WOODROW , LACHLAN , , SCHMERL SHELAH ... , ,

22 . GENERAL IN , CATEGORICAL FOR W - EXPANSIONS COMPLETE CANNOT ONE SCHEMA ) ( 's LECTURE EVAN

24 . TF ( 20171 EVANS N , , . RELATIONS L WITH LANGUAGE FUNCTIONS PARTIAL AND HUBIEKA . AMALGAMATION FREE K A BE LET CLASS . 5£ THE CLASS OF ALL THEN K FROM STRUCTURES ORDERED CLASS A RAMSEY IS . c- THMLT ( HUBIEKA 20167 , N . LANGUAGE FINITE BE A L LET RE f SET BE A CONTAINING LET , STRUCTURES CONNECTED L - FINITE OF . EQUIVALENT FOLLOWING ARE THE : THEN F ) PRECOMPACT FORBH ( @ HAS WITH EXPANSION RAMSEY PROPERTY EXPANSION . FORB .h( CATEGORICAL F) W HAS - � 2 � UNIVERSAL STRUCTURE . ' FAMILYF REGULAR EXISTS THERE � 3 � ' ) FoRBµ( F) =Fab( F SUCH THAT .

25 duumvirate ' K COUNTABLE OF CLASS A At K E is STRUCTURES . TAPE K EVERY IF UNIVERSAL Dt EMBEDDS TO • ' PACH KOMJA TH HENSON RADO , , , , SHI LAH CHERLIN SHE MEKLER , , , , @ d A EXISTENCE UNIVERSAL OBJECT OF CLASS FOR RAMSEY TEST Is THE - E HAS A FINITE CLASS A WHEN ? U UNIVERSAL OBJECT HOM - FORB ( f ) F FOR FINITE , IFF TREES OF SET FINITE F A ) TARDIF ( N . ,