Mathematisches Forschungsinstitut Oberwolfach

MATHEMATISCHES FORSCHUNGSINSTITUT OBERWOLFACH

Tag u n 9 s b e r i c ht ")1/1984

Graphentheorie

8.7. bis 14.7.1984

Die Tagung fand unter der Leitung von' Herrn' G. Ringel (Santa Cruz, California) und Herrn W. Mader (Hannover) statt.

Mehr als 40 Teilnehmer aus 13 Ländern (Australien, 'VR " China, CSSR, Dänemark, England, Frankreich, ISI;'ael,.·· Jugoslawien, Kanada, Niederlande, Ungarn, USA) haben über ihre n~uesten Ergebnisse in verschiedenen Gebieten der Graphentheorie b~richtet. In einer von Herrn P. Er~' dös geleiteten Problemsitzung wurden offene Probleme' vorgestellt und diskutiert.

Im besonderen wurden Resultate aus den folgenden Teil- ~ gebieten der Graphentheorie vorgetragen:

Färbungsprobleme, (Kanten- bzw. Knotenfärbunge~/·Larid~· ka.rten) , zusammennangsprobleme, Unendliche. Graphen, Einbettungen.von Graphen in verschiedene Fiächen, (Mini malbasen, Kantenkreuzungen), Partitionen (Turan-Graphen, Ramsey-Theorie usw.), . ~. .. '\ Wege und Kreise in Graphen.

Darüber hinaus wurden graphentheoretische Methoden be-

Als spezielle Ergebnisse seien noch die Fortschritte im "Oberwolfacher Problem" erwähnt sowie die Lösung der seit 1890 offenen Imperium-Vermutung von Heawood. Die Grundidee zu dieser Lösung ist in der abgebildeten Land karte mit 19 paarweise benachbarten, aus jeweils 4 ge trennten Ländern bestehenden Staaten angedeutet.

vortragsauszüge-

R. AHARONI: Matchings in infinite graphs

A criterion is present~d for a graph· at" any cardinality to passess a perfeet matc~ing.

B. ALSPACH: Same new results on. the Oberwolfach problem

Let F(11,12, •.• ,lr) IG denate an isomorphi~ factorization of G- into 2-factors each of which is compased of cycles af lengths l1,12, ••• ,lr. If a~l the li'5 are the same value 1, write F(l»)G instead. The results mentioned in this talk are'the following. Thm.1. F(2m) IK2~-I for all positive -integers r~1 and ~2 where I 'de'notes a·. 1-factor of K2~. Thm.2 •. If ~i~4 and even for each i, r th~n F(11.,12' .•• 'lr)IK4m,4~ for every m'~ 1 with L 1. Sm. i=1 1

Thm.·3. odd and n odd, then F .(d~_) 1K If _F(s) IKn , ~ dn for all odd d. ~(s) aso long as s i5 odd, s=9.~m~d3) and Cor.4. IKn n is an pdd multiple of s. The above is joint work .with Roland.Häggkvist.

T. ANDREAE : AGame of Cops and Robbers' There are two players c = cop player and r = robber player. First c places s cops at same of the vertices of a finite connected undirected graph, G. (Twa o~ mar~ c~~~ may be on the same vertex. ) Then' r "places ä. robber at some ·vertex. Thereafter the players move alternatively·.·· A move of' c corlsists of moving"some'of the cops along' edges to:adjacent vertice5. Similarly; a move of·r i5 defined. c' 'wins if he catches the robber, r wins if he'· avoids this forever". Let· c (Gl be the minimal number of cops that are sufficient to catch the robber~ AIGNER and FROMME proved that c(G)~3 if G' is planar, and that,

© - 4 - in general, c(G) can be arbitrarily high. Here it is shown that, for each finite graph H, there is a minimal a(H)EN such that c(G)~a(H) if H is not a subcontrac a(Kn)~(n-1) n~4, tion of G. Further, (n-3) for O(KS) 3, O(R ,3) 3, a(K 2, a(Wn)~rn/31+1 (W wheel = 3 = S) = n = with n rim vertices, Kn(K~) = complete graph with n vertices (minus an e~ge». Other results: 1. QUILLIOT showed c(G)~2n+3 if G has genus n,~. an algorithmic characterization of the graphs with c(G) = 1 is due to QUILLlOT and NOWAKOWSKI/WINKLER, 3. c(G)~2 if !V(G) 1~10, G # Petersen Graph.

I. BEN-ARROYO HARTMAN: Path Partitions and Packs in Digraphs

A path partititon P {P , ••• ,P } of a digraph is a = 1 m partition of its vertex set inta disjoint directed paths. A path partiti~n P is k-optimal if I: mln{ IP',I ,k} is as P.EP· l. l. small as possible; IPil is the number of vertices in k Pi. A partial k-colouring C is a collection of k dis-, joint independent sets. Berge conjectured the following: Let G be a directed graph and let k be a positive integer. Then for every k-optimal path partition. P there k exists a partial k-colouring C such that each path Pi in P meets min{IPil,k} different colour classes k of c • We report on recent progress on the problem and on a dual analogue, interchanging the roles of paths and colour classes.

R. BODENDIEK: Uber ebene Graphen

Ein endlicher, ungerichteter, schlichter und in die Kugel Fo einbettbarer Graph" G = (E,K) ist dadurch ausgezeichnet, daß er höchstens einmal gesä~tigt werden kann und daß der Grenzgraph G(L) für jedes Land L der ebenen Landkarte dag~gen (G,Fo-G) ein Husimibaum ist. Ist F * Fo eine orientierbare oder nichtorientierbare Fläche, so lassen sich stets 'in F e1nbettbare Graphen G finden, die min destens zweimal gesättigt werden können oder bei denen

Länder in der Landkarte (G,F-G) existieren, deren Grenz graphen keine Husimibäume·sind. Betrachtet man dagegen die Spindelfläehen o(Cn ), bei.denen der äußere Graph G ein (n~3) jeder'der'i~eren Kreis Cn ist und Graphen -G" 'G2 , ~o Spindel~ ..• , Go ein ,K 2 , kann man zeigen, daß diese flächen für sogenannte normale Graphen, die sich überdies

in o(Cn ) so einbetten lassen, daß die' n singulären Punkte von ~(~n) Ecken des eingebetteten Graphen'sind; die gleiche Sättigungs- und Landkarteneigenschaft wie Fa j~doch besitzen. Im Gegensatz zu X(Fo ) = 4 ist die chro-' matische Zahl X(O(~n» für jedes n.?: 3 gleich 5.

A. BONDY: Parity Theorems for Paths and Cycles

, , , This is areport of joint work with F.Y.' Halberstam. A gra~h is ~ (resp. odd) if every vertex has even (r~s~. od~) d~ gree. Using a technique of A.G. Thomason, we prove the fol-' 'lowing results. Theorem 1. Le,t , Pi (u) ,be the number of paths }?f length .i with initial vertex u. Then (a~ if G is even, .~~(~) is even (i~1), (~) if G 1s odd, Pi(u) is even~(i,?:2). Theorem ,2. Let Pi be the n~er of paths of leng~~ i . . - Then (a) if G is even and bipartite, Pi. is eve~ (i~1), (h) if G 1s odd and bipartite, Pi is even (i~3). Theorem 3. Let v be the.number 'of vertices' and p the number of paths. Then (a) if G 1s even, p = v(mod2), , 1 (b) if G 1s odd, P = 2v(mod2) • Theorem 1 has ari application to the question of exlstence" of generaliz~d friendship graphs (th?se graphs with the' '" • property that, for a given k,?:3, there is a~unique (u,v)-pa,th of length k for all pairs u,v of d1s.tinct vertices)':' A. Kotzig has conjectured that no such graphexists. Theorem 1 (a) imp~1es that each such graph i~ ?f o~~ order.

A. BaUCHET: Isotropie systems. Recognizing circle grap~s

Let V be a finit~ set. We consider a, 2-dim.ension~l·.yec::tor space K over GF(2) with the non-null alternate bilinear " .. ", . '. 'V'; form (x,y) ....xy and the symplectic structure induced on K

by the alternate bilinear form (a,ß)~rVEVa(v)ß(v). An iso tropie system tL,V) is defined by a totally isotropie sub V spaee cL ~ K of .dimension lVI. Isotropie systems ean be assoeiated to 4-regular graphs ealled graphie systems - and also to pairs of dual binary matroids. In.ge?eral isotropie systems unify some proper ties of 4-regular graphs and binary matroids. The application which we develop here comes from the facts that eaeh isotropie system ean be identified to a simple graph defined up to Ioeal eompiementations and t~at ~he cireie graphs (defined by assQeiating vertiees to chords of • a cireie and joining vertices iff the corresponding chords intersect) correspond so to graphie systems. A theory of 3-connectivity for isotropie systems similar to Tutte's theory for matroids allows to derive an efficient algorithm for recognizing circle graphs.

P. A. CATLIN: Homomorphisms into add eycles

A harnomorphis~ 8: G ~ H is a function fram a graph - G into a graph H such that x - y irnplies Sex) ~ 0(y) , where denotes adjaceney. We eonsider hornomorphism~ as ä generalization of the coneept of vertex coloririg (the case with H eomplete). There is a construetive characterization of the edge-minimal series-parallel graphs having no homomorphism inta eS .

For the ease H = C + ' there are variaus. recent results, 2k 1 including a sufficient condition for a homomorphism onto an odd cycle to ~e unique. • G. CHEN: Spanning balanced bipartite subgraphs of regular graphs - On R. HÄGGKVIST's Conjecture

This paper points out a counter-example' of R. Häggkvist's conjecture: "Let .G be an rn-regular graph with an even number of vertices, then G contains a spanning balaneed bipartite graph- B where each vertex has degree ~[~]."

After modifying the original conjecture, we have proved. that for m ~ 3 the modified conjeeture is true.

D. CVETKOVIC: Inte'ractive prograrnming system "Graph" an expert system for graph 'Lheory

Interactive programming system "Graph" has recently been implement~d at Faculty of Electrical Engine~ring, University of Belgrade. System "Graph" contain~ a c~mputarizeq.grap~ theory biblio~raphy, a subsystem for ,performing several tasks on particular graphs, and a theorem prover. The purpose of the system is to support research in graph theory and applications. Userls manual, a protocol of a conversa tion with the system "Graph" and other material on the system will be.presented to interested persons. w. DEUBER: Toroidal graphs admit a 5-flow Tuttels conjecture: For every finite graph G without bridges there exists an orientation G of the edges and a flow (Kirchhoff flow) lP: E(.G) .... {1,2,3,4.1 . THEOREM (M. MÖLLER): Tuttels conjecture holds for graphs of genus 1 .

A. FRANK: Edge-disjoint paths in planar graphs'

The following theorem is presented. THEOREM.ln a planar graph G k pairs of terminals on the boundary are specified. Every node not on the boundary has even degre~. ~hen ther~ exist ,k' e~ge-disjoint'p~~hs in G connecting the corresponding'terminals iff

L'surplus (Ci) ~! ~amily ~ ~-J for eyery. C1 ,C2 ,•·• ,Cl.} of _. 1 IV I cuts, where q denotes the ~urnber of odd components in G-C -C .••C . 1 2 1 (The surplus of a cut C is the difference between the number of edges in C and the n~er 9~ terminal pai~s separated by c. A set X ~ V is odd i~ the n~er of edges le~ving X and the number of terminals in X have different parity.) . . '. Th~s ~~orem is ~ common general~zation of earlier results of Okamura-Seymour and of the author.

C. GODSIL: Enumerating Latin Rectangles

Working jointly with Brendan McKay (Canberra), I have derived an asymptotic formula for the number of kxn 6 7 Latin rectangles, valid for k = o(n / ). The pr~vious best result was accurate only for k = o(n1/ 2 ). The calculation is based on an estimate for the number of ways of extending a kxn Latin rectangle R to a (k+1)xn rectangle. In fact this number is

Here G is a subgraph of Kn,n determined by R -and n n k r(G,K) ~ (_1)kp (G,k)K - k=1 where p(G,k) is the number of rnatchings in G with k edges. This leads to an estimate of the number of exten sions in terms of n,k of certain small subgraphs

... ) of G.

With some difficulty this estimate can be averaged over all kxn rectangles R, thi~ leading to the final result.

M. GRöTSCHEL: Eulerian SUbgraphs, Cuts and Certain . Binary Matroids

In this talk we study the convex hull of the incidence vectors of the cycles of a binary matroid. We prove that a descriptionof the facetsof this polytape can be ob tainded from a description of the facetsthat contain any vertex. A compl~te and nonredundant description of this polytope by linear equations and inequalities i8 given for those binary mat~oids with no F *, R or M(K )* 7 10 S minor. This implies a convenient characterization of the convex hull of the incidence vectors of Eulerian sub~ graphs of a graph and of the convex hull of the incidence vectors of the cuts in a graph not contractible to K • S

I. GUTMAN: Graphie polynomials whose Zeros are Real .. It is weIl known that the eharacteristie polynomial

' 137-144.) Let the graph G have n vert~ees V 1 ,V2 '·· .. ,vn • Let A ,A , ••• ,A be no~-negative real' numbers • 1 2 n .' "'n (~) Then the polynomia1s

R. K. GUY: Outerthickness & Outercoarseness of GraJ?hs ._.

A graph is planar.if it can be imbedded iQ t~e plane (ar sphere) •. The eomplement of the graph in. such an imbed ding is a collection of open polygons, whose boundary eon sists of v~rtiees joined by edges. I~ there i~,an,i~~d ding with all the vertices of th.e .graJ?h qn the boundary. ~.f ..( a single polygon, the graph is said to be outerplanar. ~ .

ehar~cterization of ou't:erplana:r:. graphs. is tha.,t. ~hey.. d9.. ". ..'';' not contain a subgraph homeomorphic to K4 or :K.: 2 , 3 • ':.' The thickness (resp. coarseness) of a graph is the minimum (resp. maxi.mum) nUinber of' pl'anar (resp~' non-planar) graphs into whieh the edges of the graph can be parti tioned. Outerthiekness and otitercoarseness. ~re .corr·esP9.~~.:-~~··· dingly defined by edge-partitions into (non)outerplanar ~ •• p ~ ....,• ; ~ ,-. 7" .•• i k ~. ; graphs. We g.iv~ tp.e .outerthickness and. o~terco_~rse~es.s of the usual families of graphs: Kn , Km, n' and the·- ... ,'. . d-diminsional cube.

E. GYöRI: Edge disjoint comp1ete subgraphs

One of the elassieal theorems in graph theory is Turan's theorem. We prove some theorems' about deeompositions of graphs into edge disjoint eomplete subgraphs whieh sh~w the stronger extremality of Turan graphs. E.g. we prove that any graph of n vertiees can be deeomposed into Kk'S and edges so that the s·um of the orders of these eomplete subgraphs is no more than the surn of the degrees of the k-1 partite Turan graph, providing that k is at least, 4 . We also give esti mates for the number of the edge disjoint triangles in a graph of n vertiees with m edges.

A. HAJNAL (jointly with P. Erdös): On 3-partitions of a set

We give two results relevant· to the Ramsey funetions R (n,n) and R (n,n,n). It is weIl known that 3 3 c n 2 c,n 2 2? < R (n,n) < 2 3 for sorne eo,e, > o. The'next theorem'says that the lower boun~ can not be improved using "justly distributed" parti tions ... ~.> ~ Theorem 1.' 3 3c a > 0 \J-m V(Ko,K,) ([rn],3 =', Ko U·K, :::;0 3H c: In 3i < 2

I H I ~ Ca' Vlogm "I [H] 3 n i< i I > a (I ~ I))., 2 n logn Theorem 2. R (n,n,n) -> 210g.1ogn for n > n • 3 o We conjecture that' Theorem 1 extends to' every a < i. For history and ear1ier results see "Combinatorial Set Theory" by Erdös, Hajnal, Mate, and Rado.

H. HARBORTH: Multiple crossings in drawings of graphs

Realizations of a graph in the plane are considered where two lines have at most one point in common, either an end point or a crossinq. For graphs with 2m vertices at most rn-feld crossings are possible. It is .proved, that the maxi mum number ?f m-fold crossings is 2 for m = 3 and m = 4, and at least 2 in general.

A.J.W. HILTON: Edge-colouring of graphs

.~hi~. 1s joint with myself and A.G. Chetwy.nd. Let x' (G) and. 6(G) denote the ~dge-chro~atic number and maximum degree of a simple graph G. If IE(G) I. > A(G)lIV~~) IJ., tpen ca~l G an, overfull graph. ~t i~ well-known that if G ~s an,ove.rfull graph then ~'(G) = .6(G)+1., Theorem 1. Let G have r vertices of maximum.. degree, let 7 IV(G)I' 2n or 2n+1 and l~t a(G) ~ n + 2' r - 3. If Xl (G) A(G)+1 than it follows that either G 15 overfull er G .has an edge-cut. ~ w~th 181 < r - 2 . such that

~(G) G ...... 8 =:= G" UG2' where G 1 n G2 .= cf>, .l~ (G 1 ) and G1 i5 overfull. Theorem 2. Let I.V(G) I be even and let G be regular of 6 degree d:'(G), satisfying d(G) ~"7 )V(G) 1.. Then .x l (G,.) Ö (G) •

L. LOVA8Z: Independent 's'ets in claw-free graphs

A claw. in a graph 15 .an .induced .K~1,"3. The class of. claw-free graphs. includes all lihe-graphs',' and, many non-l~ne graphs, e.g.

The proble~ of, find~ng a ~~ximum-i~dependent set of' points in a claw-fre~ gr~~h includes the ..~el+~solved matching prob~.em. Polynomi~l.time algorithm~__ ~or this problem w~re .givel?- by '. Sbihi and Minty. In this t~~k it 1s s~o~ that every .claw-~ree graph arises from a line-graph by gluing on pieces with inde-

pendence number ~ 2 in a well-described way. This gives use to an algorithm which reduces the independence number problem for claw-free graphs to a matching problem.

W~ MADER: On minimally n-connected digraphs

Für einen minimal n-fach zusammenhängenden, gerichteten Gra phen D sei 0 der von den Kanten (x,y) mit y+(x:O) > n und 0 Y-(YiD) > n erzeugte Teilgraph, wobei y+(x:O) bzw. y-(x;D) den AUßengrad bzw. Innengrad von x in D bedeute. Es wird ge- 4It zeigt, daß 0 keinen alternierenden .Zyklus enthält. Dies ist 0 äquivalent dazu, daß ein 0 zugeordneter paarer-Graph keinen 0 Kreis enthält. Hieraus ergeben sich ähnliche Resultate wie im ungerichteten Fall. (1) Für jeden endlichen, minimal n-fach zusammenhängenden, gerichteten Graphen 0 -gelten !{XEE(D): "Y+ (x; D) =n} I ~ n lind I{xEE (0) : y + (x; D) =n} I + !{xEE (D) : y - (X-i D) =n} I• ~ ~~~1 2101 + ~n-1 • (2) Für jeden endlichen, minimal n-f~ch zusammenhängenden, gerichteten Graphen 0 gilt für die Kanten- zahl 1101 I ~ 2nlDI - n(n+1) und im Falle IDI > 4n + 5 so- gar 1101 I ~ 2n (IOI-n), wobei die D charakterisiert werden, für welche das Gleichheits~eichen gilt.

I. MENGERSEN: Eine Abschätzung für die Ramseyzahl r(KS-x)

Unter der Ramseyzahl r(G) eines Graphen G versteht man die kleinste natürliche Zahl- p, so daß-bei jeder 2-Färbung

der Kanten des Kp ein einfarbiger Teilgraph G vorkommt. ~ Bislang ist r(Kn ) nur für n 4 exakt bekannt, für ~ ~ r(KS ) weiß man nur 42 r(KS) 55 . Als eine Art An näherung an r(KS ) ist r(K5-x) von Interesse (KS-X ent ~iner steht aus KS durch Entfernen Kante). Die beste bis her bekannte obere Schranke für r(K5-x), 24, wird zu 23 verbessert. (Die beste bisher be~annte untere Schranke ist 21.)

H. MEYNIEL: Partitions of Digraphs into Paths or Circuits

An upper bound on the minimum number Of paths or circuits in a partition of the ares of a digraph arid same conjec tutes related to this topic are given.

R. H. MÖHRING: Two Theorems on Graph Substitution

Suppose we want t~ represent the maximum clique weight wG(xf of G as a function of the'node weights x = (XVIVEV) in a decomposed way as [f1(xvlv~V1) wG(x) f , ••• ,fm(xvlve:vm)]

f,f".~.,fm where V" ... 'Vm form a partition of V, and are realvalued functions. 'Under certain conditions on.the 2 .?ijec~iy~ly f i (no f i may map a subset homeomorphic to R into Rl ) we show that G must then decompose according to graph substitution (= X-join) as

G = G'[G'lv1, ••• ,Glvm],

where GIVi is the subgraph of G induced by Vi and G' is the'quotient graph of G with respect to the partition {Vl,···,Vm}· Appl~g results on the asymptotic relative frequency of (subs~itu~ion-) indec~mposable partia~ orde~sl we .then ~~ow that almost all comparability graphs are uniquely parti~l~y, orderable (UPO), i.e.

lim # UPO comp. graphs with n points = # comp. graphs with n points

As a conseque~ce, the number of compa~ability.graphs.is asymtotically equal to half the number' of.parti~l orde~~.

H. M. MULDER: Distance-hereditary Graphs

Distance-hereditary graphs are connected g~aphs. ~I.l ~hich,. ," all induced graphs are i~ometric (i.e. all'induc~d paths:. are geodesics). Examples of such graphs are provided ~y ,-. complete, multipartite graphs and ptolemaic graphs. Every finite distaJ)ce-hereditary graph can,be, obtained . . from K, by iterating the following two. opera~ion~: , adding pendant vertices and vert~x splittings. Using'. this~.· result characte~izations of (infinit~) distance-heredit~ry graphs can be deduced: in terms of the distance function d

(involving some four-point-condition), or in terms of the interval funetion I of the graph, or via forbidden iso metrie subgraphs. Related results on ptolemaie graphs and parity graphs are given.

J. PACH: On universal graphs

A elass ~ of infinite grap~s is said to have a univeral element Go E 9f if G c Go holds for every GE: CJ · Theorem 1. The class of countable planar graphs does not have a universal element. Theorem 2. (P. Komjath & J.P.) Assume GCH. Let < a ~ ß ~ Y be cardinals, a < w ~ Y Then the elass lJy(Ka,ß) of all Ka,e-free graphs on y vertiees has. a universal element iff

(i) y > w or

(ii) y = w a = and ß ~ 3 •

The special.eases (a=y>w,ß=1) and (a=ß=2,y=w) of Theorem 2 were p!oved by Shelah (1973) and' Hajnal-Pach (1981), Theorem answers a question of Ulam.

K.-P. PODEWSKI: Critical families

Let F = (F(i) liEI) beafamily of sets. F is called critieal if F has an injeetive ehoieefunction and f[1] = U{F(i) liEI} for every injective choieefunction. Critical families can be defined by transfinite recursion on Cl as foliows. If a = .Ua , then F E Ca iff there is a chain (I ) ß

G. RINGEL: Beweis der, ebenen I-mperium-Vermutung von,Heawood

Es wird bewiesen, daß in der Ebene eine Landkarte existiert mit Gm paarweise benachbarten Staaten, wobei jeder Staat aus m getrennten Ländern besteht (m~2). Dies war bisher nur für' m=2 von Heawood "'890 und für m=3 oder 4 von Taylor 1980 be wiesen.

Y. RODITTY: Packing and covering of complete graphs by trees

It is shown that: [~~~=~n gl~:~l, p(n,Tk } = and C(n,Tk } = n > 9 , where T k is _.any tree of order k ~ 6,,' P(n,G) the maximal number of pairwise,edge disjoint graphs G in the complete graph K n , and C(n,G) the minimum number of graphs G whose union is Kn • ([xl de notes 'the largest integer not exceeding x and {x} the least integer,not less than x.) It was' as~ed ~he ~ollowing: I) Is it true that for each tree T of order k > 6: k P (n,T ) = (n-1)] and C (n, ,T ) {n (n-') 1 k [n2(k-l) k 2(k-1)J (no som~constant to be determined) .

11) 15' it true that for' each integer' k > 6, there,exist integers' .m,'n" such that' if (k-1) I mn then K 'is m,n deeomposed inta isomorphie eopies of each tree T of' k order k? Ex: For k G I K4 ,5 is decomposed into each tree of order 6.

V. RÖDL (joint' result with Peter Frankl): 'Oensity theorems for'graphs and hypergraphs

Let G: (n.,e.), i = 1,2, ... , n~ ~ m be a sequence of 1. ~ 1. .... r~uniform edges~ hypergraphs of n i vertiees and e i The density of this sequence is a, '0 ~ a < 1, if a is the largest'real number for whieh there"is'a subgraph G~, G~ ~.'X) (mi,e'i) of '(ni,ei ) for whieh mi and lim e!/(m/) = a • The theorem af Erdös, Stone and Simonovits 1 .

states that for r =_ 2 the only possible values of the densities are 1 and - .! where t t = 1 ,2, ... It was conjectured by P. Erdös that for r > 3 the set Dr c [0,1] of possible densities forms a weIl ordered set. We disprove this conjecture and show the following

Theorem: For every E > 0, r ~ 3 there exists l(r) such that for any 1 > l(r)

1 - 1 < d < 1 - 1 + E i r - 1 rr-1 for some d EDr

I. SCHÖNHEIM: Application of small Turan numbers

Let f: E(K ) ~ E(K ). The K A is caiied free if n n 4 fee) E E(A) for eE E(A). Theorem 1. If fee) * ethen the smailest number enforcing a free K4 for every f is 10. Theorem 2. The smallest number enforcing either an edgewise. fixed triangle or a free triangle is 13.