A note on Turán number Tk ( 1, kn, )

Transcription

1 A oe o Turá umber T (,, ) L A-Pg Beg 00085, P.R. Cha apl000@sa.com Absrac: Turá umber s oe of prmary opcs he combaorcs of fe ses, hs paper, we wll prese a ew upper boud for Turá umber T (,, ).

2 . Iroduco Suppose ha S s a se of elemes, for a o-egave eger r, as usual, he se of all he r S S r sad for subses of S. Suppose ha s a o-egave eger, r, s a famly of subses of S, s called a (, r) Turá famly f for ay S X, here s a member A such ha A X. Deoed by Tr (,, ) he mmal r sze of (, r) Turá famles, he umber s also called Turá umber, whch frs proposed P. Turá s papers [5], [6]. For Tr (,, ), here are he esmaos ad r Tr (,, ), (.) d S Tr (,, ) (.2) where r d, S, d or S d, so for 3 Tr (,, ) d. (.3) The deals for he resuls above refer o see papers [2], [4]~[6]. The resul (.3) wll be rval he case d,.e. he case r 2, for sace, r. For T (,, ), K.H. Km ad F.W. Roush [3] proved ha T (,, ), (.4) / 2log 2log V.Rödl ad P. Fral [] mproved ha log T (,, ). (.5) /loglog I hs paper, our ma resul s ha.

3 Theorem. where log T x c0 ( ) ( e ) dx, log c ( ) c ( ) C 0 (,, ), ( ) ( ) C /, c 0 0 c ( ), c ( ) 0. (.6) 2. Proof of Theorem As usual, 0 sad for he se of all o-egave egers, suppose ha s, ad h are hree posve egers, deoed by s v v( x, x2,, x) 0, x s, Vsh (,, ) vv, xh,, Vsh (,, ) vv,max x h. s s (2.) Defe ( sh,, ) ( x ), ( sh,, ) ( x ). (2.2) ( x,, x ) V( s,, h) ( x,, x ) V( s,, h) Lemma. CC s ( h ) h C C s ( h ). (2.3) ( sh,, ) ( ) ( ) s Proof. Suppose ha gx ( ) ( sh,, ) x s he geerag fuco of ( sh,, ) for he s h h parameer s. Deoed by h h, le p( x) ( x ) /( x), p ( x) ( x ) /( x), he has gx p x p x ( ) ( ) ( ) x ( h ) x hx h h h ( ) ( C hxc hc ) x C x ( h) x ( ) CC hc C s s sh sh C x.e. CC s h h C C s h. ( sh,, ) ( ) Le s, ad assumed ha.

4 Lemma 2. v 2 log 2 2 vlog v o v (2.4) Proof. From he calculus we ow v v v v v log dx log log dx log log x x ad v v log dx dx x x v v v vlog v vlog 0 ( ) vlog v o( v ). Lemma 3. d h (, sh, ) C ( / ) ( / ) ( / ) o C ( ) C h h h h h / 2 () h 2 3 ( ( / )) (( / ) ( / ) ) (2.5) Proof. By (2.3) ad (2.4), has (, sh, ) ( )( C C hc C ) h h h h h C ( ) C C 0 0 h C ( / ) (( )/( )) (( )/( )) h 2 3 / 2 o() C ( ) C( ( / )) (( h/ ) ( h/ ) ) h h h h C ( / ) ( / ) ( / ) h h h h 2 3 o C ( ) C h h / 2 () ( ( / )) (( / ) ( / ) ).

5 x Lemma 4. Le () ( e ) dx, he () log() c () c (). (2.6) 0 where log x c0() ( e ) dx, () ( ) C /, 0 c c ( ), for. Ad le Proof. x x () ( e ) xe dx, he ( ) (). (2.7) log( ) x x x ( e ) dx ( e ) dx ( e ) dx log( ) ( ) C log( ) c0 ( ) /. 0 Ad x x x () ( ) ( e ) x ( e ) xe dx ( ) ( e ) ( e ) ( ) (). Lemma 5. ( sh,, ) ( ( )) C. (2.8) s/ h Proof. Wh (2.5), (2.6) ad (2.7), has s/ h (, sh, ) / h d h C ( / ) ( / ) ( / ) h h h h 2 3 o C ( ) C h h / 2 () ( ( / )) (( / ) ( / ) ) / h h C ( e ) ( e )) e h/ h/ h/ / h / dh

6 C ( ) C h 2 3 / 2 o() ( ( / )) (( h/ ) ( h/ ) ) / h x/ x/ x/ ( e ) dx ( e ) xe dx / / C o() C 3 ( ) C e 2 3 ( ) / ( ) / x x x C ( e ) dx ( e ) xe dx 3 o() C ( ( e ) ) 2 3 ( ( )) C C ( ( )). Lemma 6. h( s,, h) ( ( )) C. (2.9) s/ h Proof. I s clear ha ( sdh,, ) ( sdh,, ) ( sdh,, ). So, h( sh,, ) h( ( sh,, ) ( sh,, )) s/ h s/ h ( ) C ( s,, h) s/ h( ) ( ( )) C. The Proof of Theorem. For wo o-egave egers a ad b, a b, deoed by [ ab, ] { aa,,, b}, ( ab, ) { a, a2,, b}, so, he case b a, ( ab, ). Assumed ha S 0,,,, for a subse X S, X { x0, x,, x r }, x0 x2 x r. Deoed by ( x, x ), 0,,, r, ad ( x, ), ae X as a subse of r r

7 he complee resdue sysem ha 0,,,, mod( ), he may be decomposed o [0, ] { } { } { }, x0 0 x x2 2 x r r wh a possble roao, may be assumed ha x 0 0. Le ( X ) S \ X. Deoed by b, 0 r, of he larges gap of se X he crcle. ad h maxb X ( X ) ( X ), clearly, 0 r, ha s, X 0 r For a o-egave eger, 0, ad for a subse X S, defe h s he sze we se ( X ) x mod( ). xx S L X X, ( X) {0,,, hx }. I s o dffcul o demosrae ha L s a (, ) Turá famly: Suppose ha Z s a ( ) -subse of S. If ( Z) Z, ae X Z \, he ( X ) 0, ha s, X L ; f ( Z) ( Z), ae X Z \{ x }, he ( X ) x b hx,.e. X L. Besdes, we ow ha each -subse X of S occurs exacly hx L s, hece L 0 X S T (,, ) h ( ) h( s,, h) C. h X We have made hree lss for he precse values of umbers c 0 ( ), c ( ) ad L compuer, whch are aached he ed of he paper as a appedx. wh ad of

8 Refereces [] P. Fral, V.Rödl, Lower bouds of Turá s problem, Graphs& Combaorcs, (985), [2] G. Kaoa, T. Nemez, M. Smoovs, O a graph-problem of Turá. Ma Lapo 5(964), [3] K.H. Km, F.W. Rouch, O a problem of Turá, Sudes pure mah. (ed: P. Erdös), Brhauser, 983. [4] A.P. L, Four Exremal Problems combaorcs of fe ses, dsserao, Is. of Appled Mah, Acad. of Chese Sc [5] P. Turá, A exremal problem graph heory, Ma. Fz. Lapo 48(94), [6] T. Turá, Research problem, Maguar Tud. Aad. Ma. Kuao I. Kozl. 6(96),

9 Appedx c ( ) c ( ) c ( ) c ( ) Ls. c ( ) c ( ) c ( ) c ( ) Ls 2. Lm L L/ C ( ( )) /( ) / / / /2.860 / / / / / / / / / / / / /

10 L m m L, L L, 32, ( ) log( ) c( ) c0( ). 0 0 Ls 3.