Non-uniform Turán-type problems
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1 Joural of Combatoral Theory, Seres A wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at Chcago, Chcago, IL 60607, USA Receved 9 Jue 2004 Avalable ole 12 Jauary 2005 Abstract Gve postve tegers,, t, wth 2, ad t<2, let m,, t be the mmum sze of a famly F of oempty dstct subsets of [] such that every -subset of [] cotas at least t members of F, ad every 1-subset of [] cotas at most t 1 members of F For fxed ad t, we determe the order of magtude of m,, t We also cosder related Turá umbers T r,, t ad T r,, t, where T r,, t T r,, t deotes the mmum sze of a famly [] r [] r F F such that every -subset of [] cotas at least t members of F We prove that T r,, t = 1 + o1t r,, t for fxed r,, t wth t r ad 2004 Elsever Ic All rghts reserved MSC: 05C65; 05D05 Keywords: Hypergraph Turá problem; Extremal set theory 1 Itroducto Gve postve tegers,, t, wth 2 ad t<2 We call a famly F 2 [] \ a, t-system f every -subset of [] cotas at least t sets from F, ad every 1- subset of [] cotas at most t 1 sets from F Aalogously, gve tegers,, t, r, wth 1 r ad 0 t <2,aTurá- r,, t-system Turá- r,, t-system s E-mal address: mubay@mathucedu D Mubay 1 Research supported part by NSF Grats DMS ad DMS Research supported part by NSF Grat DMS , a VIGRE Postdoctoral Fellowshp at Uversty of Illos at Chcago $ - see frot matter 2004 Elsever Ic All rghts reserved do:101016cta
2 D Mubay, Y Zhao Joural of Combatoral Theory, Seres A a famly F [] r F [] r so that every -subset of [] cotas at least t members of F We deote by m,, t the mmum sze of a, t-system, ad by T r,, t T r,, t the mmum sze of a Turá- r,, t-system Turá- r,, t-system Computer scetsts troduced ad studed m,, t see [4,5,7] for ts hstory ad applcatos Sloa et al [7] proves that m,, t = Θ for 1 <t<ad m, 3, 2 = , ad Füred et al [4] proves that for fxed, m,, 2 = 1+o1T,, 2 T r,, t especally T r,, 1 s well ow ad sometmes called the geeralzed Turá umber, though ts ouform verso T r,, t appears ot to have bee studed before Note that whe T r,, t = Ω r for fxed t,r <, the asymptotcs of T r,, t are ot ow for ay r 3 ad t 1 see [1] for a troducto to ths problem, ad [3,6] for surveys the case t = 1 I ths ote, we frst study m,, t for all 1 t <2, determg ts order of magtude for fxed, t Theorem 1 Let 2, 2 t <2, ad 1 <t The there exsts a costat c, depedg oly o ad t, such that c m,, t for t 1 m,, t for t > Remar Whe = 1, yelds m,, t = Θ for 2 t 1, a result from [7] We ca obta the exact value of m,, t for some choces of t Fort = 1,, ad 2 1, t s trval to see that m,, t s equal to,, ad, respectvely We clam that m,, 2 2 = To see ths, we call a, t-system H mmal f S H S S H S for every, t-system H IfF s a mmal, t-system for t = 2 2 2, ad A F, the 2 A \ F, sce replacg A by B A for some B F creates aother, t-system that cotradcts the mmalty of F Cosequetly F =, because S F ow mples that F \ S s a, t-system Sce t = 2 2, we must have F = [] Before proceedg our upcomg Theorem 3 whch relates T r,, t ad T r,, t, we mae the followg observato Observato 2 Let 1 r ad 0 t < Let be the uque teger satsfyg 1 t < ad let t0 = t 1 0 If F s a Turá,, t 0 -system, the F = 1 [] F s a Turá- r,, t-system Ths mples that T r,, t 1 + T,, t 0 Theorem 3 Let r,, t,, t 0 be fxed as Observato 2 1 If t 0 = 0, the T r,, t = 1 2 If t 0 1, the T r,, t = 1 + o1 1 + T,, t 0
3 108 D Mubay, Y Zhao Joural of Combatoral Theory, Seres A Coecture 4 Gve r,, t,, t 0 as Observato 2, T r,, t = 1,, t 0 + T Most of our otatos are stadard: Gve a set X ad a teger a, let X a ={S X : S =a}, X a ={S X : 1 S a}, X a ={S X : S a}, ad 2 X ={S : ad Ft = [] t \ Ft Let F t = t F ad F t = t F Wrte FX for F 2 X Ar-graph o X s a hypergraph F X r S X} ForF 2 [], let F t = F [] t 2 Proofs Proof of Theorem 1 The theorem follows easly from the followg four statemets 1 If 1 <t, the m,, t 2 If <t, the m,, t 3 If t> 1, the m,, t 4 If 1 <t, the m,, t c, where c depeds oly o ad The proofs of 1 ad 3 are straghtforward, so we oly prove 2 ad 4 Proof of 2 Cosder the smallest [1,] ad the largest [1,] such that 1 + l= l t l= l Such, exst sce <t We frst show that Ths s trval for =, so assume that < The choce of mples that [ ] 1 1 t> = + + l + 1 l + 1 l=+1 + l=+2 l= l l=+2 1 l + 1 Sce ths s equal to + 1, the choce of mples that Now let F = [] l= l Every -set of [] has l= l t members of F; every 1-set of [] has l= l l= l t 1 members of F Cosequetly, m,, t F Proof of 4 Frst, the assumpto, t-system Let K 1 < mples that < Let F be a deote the complete -graph of order 1 The K F for all [, ], otherwse we obta a -set whch cotas t members of F, a cotradcto Recall that the Ramsey umber R s, t s the smallest N such that every -graph o N vertces cotas a copy of ether K s or K t By Ramsey s theorem, R s, t s fte Defe m 2+1 =, ad m l = R l 1,m l+1 recursvely for l = 2, 2 1,,2, 1
4 D Mubay, Y Zhao Joural of Combatoral Theory, Seres A We clam that every m 1 -set of [] cotas at least oe member of F Ideed, cosder a m 1 -set S 1 Because K 1 F, the defto of m 1 mples that there exsts a m 2 -subset S 2 S 1 wth all of ts 1-subsets abset from F Repeatg ths aalyss, we fd a sequece of subsets S 3 S 2+1 = S of szes m 3 > >m 2+1 =, respectvely The -set S thus cotas o members of F of sze 1,, O the other had, the -set S must cota at least t> 1 members of F, thus at least oe member of F Hece S cotas a member of F By a easy averagg argumet, we obta F m1 = c Proof of Theorem 3 Part 1 Let F [] r be a mmal Turá- r,, t-system We are to show that F 1 Cosder F< = 1 [] \ F For every -set S of [], 1 = t FS = F < S + F S 1 = F < S + F S Therefore F < S F S Cosequetly usg x x s decreasg x for 0 x, F < < S [] F < S S [] F S F Thus F< F, ad therefore F = F < + F F < + F < = 1 The ma tool to prove the secod part of Theorem 3 s the followg well-ow fact For a famly G of r-graphs, the extremal fucto ex, G s the maxmum umber of edges a r-graph o vertces that cotas o copy of ay member of G Theorem 5 Erdős-Smoovts [2] For every ε>0 ad every famly of r-graphs G, each of whose members has vertces, there exsts δ > 0, such that every r-graph o vertces wth at least ex, G + ε r edges cotas at least δ copes of members of G Proof of Theorem 3 Part 2 It suffces to show that for every ε>0, there exsts 0 = 1 0 ε,,t>0, such that for all 0, T r,, t 1 ε + T,, t 0 I fact, ths follows from the followg clams tag 0 = max{ 1, 2 }: a T r,, t T,, t 0, b T,, t 0 >1 ε2t,, t 0 for > 1, c T,, t 0 T,, 1 > 21 ε ε 1, for >2 Sce a ad c are easy to see, we oly prove b Suppose that F s a Turá-,, t 0 - system Let G be the famly of all -graphs o vertces wth more tha t0 edges Let δ be the output of Theorem 5 for puts ε[2 ] ad G, ad choose 1 so that δ > for all > 1 ote that 1 = 1 ε,,t We wll show that F >1 ε2t,, t 0
5 110 D Mubay, Y Zhao Joural of Combatoral Theory, Seres A for > 1 Suppose, for cotradcto, that F 1 ε2t,, t 0 Sce ex, G = T,, t 0 ad T,, t 0, F 1 ε T,, t 0 2 = ex, G + ε 2 T,, t 0 ex, G + ε 2 By Theorem 5 appled wth put ε[2 ], the -graph wth vertex set [] ad edge set F cotas at least δ copes of ot ecessarly the same members of G I other words, there are at least δ -sets of [] that cota fewer tha t0 members of F Now cosder the famly of -sets of [] whch cotas at least oe member of F for some > Deote ths by K ad let K = < K Sce K F ad F, K = K F < < 1 1 F < 1 1 Sce δ > > K for > 1, at least oe -set S of [] cotas fewer tha t 0 members of F ad o member of F for > Cosequetly S cotas fewer tha t 0 members of F Ths cotradcts the assumpto that F s a Turá-,, t 0 -system Acowledgmets The authors tha Gy Turá for troducg them to the parameter m,, t, ad the referees for suggestos that mproved the presetato Refereces [1] WG Brow, P Erdős, VT Sós, Some extremal problems o r-graphs New drectos the theory of graphs, : Proceedgs of the Thrd A Arbor Coferece Uversty of Mchga, A Arbor, MI, 1971, Academc Press, New Yor, 1973, pp [2] P Erdős, M Smoovts, Supersaturated graphs ad hypergraphs, Combatorca [3] Z Füred, Turá type problems, Surveys Combatorcs, 1991 Guldford, 1991, Lodo Mathematcal Socety, Lecture Note Seres, vol 166, Cambrdge Uversty Press, Cambrdge, 1991, pp [4] Z Füred, RH Sloa, K Taata, Gy Turá, O set systems wth a threshold property, submtted for publcato [5] S Jua, Computg threshold fuctos by depth-3 threshold crcuts wth smaller thresholds of ther gates, Iform Process Lett [6] A Sdoreo, What we ow ad what we do ot ow about Turá umbers, Graphs Comb [7] RH Sloa, K Taata, Gy Turá, O frequet sets of Boolea matrces, A Math Artf Itell
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