On the ratio of maximum and minimum degree in maximal intersecting families

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1 On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting family F ( is we conside the atio R(F = (F δ(f of its maximum and minimum degee. We detemine the ode of magnitude of the function m(n,, the minimum possible value of R(F, and establish some lowe and uppe bounds on the function M(n,, the maximum possible value of R(F. To obtain constuctions that show the bounds on m(n, we use a theoem of Blokhuis on the minimum size of a non-tivial blocking set in pojective planes. Keywods: Intesecting families, maximum and minimum degee, blocking sets AMS subject classification: 05D05, 05B5 1 Intoduction A family F of sets is said to be intesecting if F 1 F holds fo any F 1, F F. In thei seminal pape, Edős, Ko and Rado showed [3] that if F is an intesecting family of -subsets of an n-element set X (we denote this by F ( ( X, then F n 1 1 povided Alféd Rényi Institute of Mathematics, P.O.B. 17, Budapest H-1364, Hungay and Eötvös Loánd Univesity, Depatment of Compute Science, H-1117 Budapest Pázmány Péte sétány 1/C. nagyzoltanloant@gmail.com. The autho was suppoted by the Hungaian National Foundation fo Scientific Reseach (OTKA, Gant no. K ozkahya@illinoisalumni.og Alféd Rényi Institute of Mathematics, P.O.B. 17, Budapest H-1364, Hungay. patkos@enyi.hu. Reseach is suppoted by OTKA Gant PD and the János Bolyai Reseach Scholaship of the Hungaian Academy of Sciences. Cental Euopean Univesity, Depatment of Mathematics and its Applications, Budapest, Nádo u. 9. H-1051, Hungay. vizemate@gmail.com. 1

2 that n. Lots of genealizations of the above theoem have been consideed eve since and lots of eseaches have been inteested in descibing how intesecting families may look like. One of the quantities concening intesecting families that has been studied [, 4] is the unbalance U(F = F (F whee (F denotes the maximum degee in F. In this pape we define anothe notion to measue the balancedness o unbalancedness of F: if δ(f denotes the minimum degee in F, then ou aim is to detemine how small and how lage R(F = (F δ(f can be. To avoid δ(f = 0 we will always assume that F = X. Also, as by consideing appopiate subfamilies one could modify the value R(F easily, we will estict ou attention to maximal intesecting families, i.e. families with the popety G ( X \ F F F F G =. Fo sake of simplicity we will also assume that the undelying set X of ou families is = 1,,..., n}. With the above notation and motivation we define ou two main functions as follows: M(n, = max R(F : m(n, = min R(F : ( F ( F is maximal intesecting with F = is maximal intesecting with F = The family giving the extemal size in the theoem of Edős, Ko and Rado seems to be a natual candidate fo achieving the value of M(n,. In fact, most families F that occu in the liteatue have R(F = Θ( n. In Section we will pove the following theoems showing that both M(n, and m(n, have diffeent ode of magnitude. Theoem 1.1. (i Fo any values of n and, we have M(n, n +. In paticula, if < log n, then M(n, (1 o(1n holds, log log n (ii if + < n, then M(n, n + 3 n + holds, in paticula if ω(1 = < ( 3 log n, then we obtain M(n, n. log log n At fist sight, the bound seems to be vey weak, but we will show in Section 3 that it cannot be stengthened too much in geneal. A tivial lowe bound on m(n, is 1. The next theoem states that n is anothe geneal lowe and we constuct families showing that this is the ode of magnitude of m(n, as long as = O(n 1/. Fo lage values of we obtain egula maximal families showing the tightness of the tivial lowe bound. Theoem 1.. (i Fo any n and, the inequality m(n, n holds, (ii if n 1/, then m(n, = Θ( n holds, (iii if ω(n 1/ = = o(n and (n/n is monotone, then thee exist infinitely many values n and = (n such that m(n, (n = 1 and holds. }, }.

3 Poofs In this section we pove Theoem 1.1 and Theoem 1.. Poof ( of Theoem 1.1. To pove (i, let us conside a maximal intesecting family F. Let us patition F into two subfamilies F1 and F whee F 1 := F F : x F such that F \ x} F fo all F F} and F = F \ F 1. Claim.1. F. Poof of Claim. Let d j denote the maximum numbe of sets in F that contain the same j-subset. Fo any j < and j-subset J that is contained in some F F thee exists an F F with J F =. Thus d j d j+1 holds. Since d = 1 and F d 1, the claim follows. Let τ denote the coveing numbe of F, i.e. the minimum size of a set meeting all sets of F. Clealy, if τ =, then F 1 = and thus by Claim.1 F and R(F. Assume τ <. We will show a oughly n to 1 mapping f fom F 1 to F min, the subfamily containing one fixed vetex y of minimum degee. Fo any F F 1 let g(f be an element of F so that F \ g(f } F fo all F F (such an element exists by definition of F 1. Let us define f(f = F if y F and f(f = F \ g(f } y if y / F. Note that f(f F as aleady F \ g(f } meets all sets in F and by assumption F is a maximal intesecting family. Obseve that at most n + 1 sets can be mapped to the same set G as all such sets should contain G \ y}. This concludes the poof of (i as R(F R(F 1 + F. To pove (ii we need to show a constuction. Let us wite S = [, ] and S 0 = [, 1] and define F 1 = 1} G : G F 3 = ( } S, F = 1 ( S, F 4 = (S \ S 0 i} : 1 i n}, F = 4 j=1f j. Claim.. The family F is maximal intesecting. ( } S 1, i} H : 1 i n, H \ S 0, Poof of Claim. F is clealy intesecting as all of its sets, except those coming fom F, meet S in at least 1 elements. A set F fom F meets any othe fom F 1 F as they both contain 1, a set fom F 3 because of the pigeon-hole pinciple, and a set fom F 4 as by definition F S S 0. To pove the maximality of F let us conside a set T / F. If T S <, then any -subset of S \ T is in F and thus T cannot be added to F. As all -subsets of S ae aleady in F, it emains to deal with the cases T S = 1 and T S =. If T S = 1, then 1 / T as those sets ae in F 1 and T S S \ S 0 as those sets ae in F 4. But then a set F fom F with F S = S \ T is disjoint fom T, thus T cannot be added to F. 3

4 Finally, if T S =, then if 1 / T, then 1} (S \ T F 1 is disjoint fom T and thus T cannot be added to F. If 1 T, then, as T / F, we must have T S = S 0. Then we can find a set disjoint fom T in F 4. All we have to obseve is that in F the degee of 1 is ( 3 1 and the degee of i fo any 1 i n is ( 3. ( + ( 3 1(n + Note that the poof of Theoem 1.1 (i gives an uppe bound M(n, n + fo any value of and n. Conjectue.3. If = o(n holds, then the ode of magnitude of M(n, is Θ(n. Now we tun ou attention to the function m(n,. In the poof of Theoem 1. we will use the following theoem by Blokhuis on blocking sets of pojective planes (fo a shot suvey on the topic see [6]. Theoem.4 (Blokhuis, [1]. Let Q be a pojective plane of ode q and B be a blocking set (a set that meets all lines of the pojective plane of size less than 3 (q + 1. If q is pime, then B contains a line of the pojective plane. We will also need the following stengthening of Chebyshev s theoem. Theoem.5 (Nagua, [5]. Fo evey intege n 5 thee exists a pime p with n p (1 + 1/5n. Poof of Theoem 1.. To pove (i we make the following two easy obsevations: fo any intesecting family F we have (F F / as fo any set F F the inequality x F d(x F holds. Also, the aveage degee in F equals F. As the aveage degee is at least as n lage as the minimum degee, we obtain R(F = (F F δ(f F n = n. Note that the poof does not use the fact that F is maximal. To pove (ii and (iii we need constuctions. Suppose fist that n 1/ holds. By Theoem.5 we can pick a pime p such that < p < (1 + 1 = 4. Let P denote a pojective plane of ode p with vetex set [p + p + 1]. Let us define the following maximal intesecting family ( } F n,,p = F : l F fo some line l P. 4

5 Note that F n,,p is intesecting as any two of its sets intesect as they both contain lines of a pojective plane and F n,,p is maximal as if G ( does not contain any line of P, then by Theoem.4 and < 3p we know that thee exists a line l in P such that l G = and this line can be extended to a set F l l such that F l G = holds. As evey vetex is contained in p + 1 lines of P we have that d(x = (p + 1 ( ( n p 1 p 1 + p n p p if x [p + p + 1], while fo any y [p + p +, n] we have d(y = (p + p + 1 ( n p p. Theefoe we obtain R(F n,p, = p( ( n p p + (p + 1 n p 1 p 1 (p + p + 1 ( n p p + 1 n p 1 p n p whee the last inequality follows fom p 4. Note that by the pime numbe theoem one 5 can pick p such that p povided that is lage enough, and thus impove the constant 3 5 to It emains to pove (iii. Conside the following geneal constuction F k,p,s ( whee 1 k is an odd intege, p is a pime, 0 s p and n = k(p + p + 1, = k+1 (p s. Fo 1 i k let P i be a pojective plane of ode p with undelying set [(i 1(p + p , i(p + p + 1] and let us define ( ( } [k] F k,p,s = F : F contains a line of P i if i I fo some I k+1. As any two lines of a pojective plane intesect each othe and so do any I, I ( [k] k+1, the family F k,p,s is intesecting and by Theoem.4 and s p we obtain the maximality of F k,p,s. As the constuction is symmetic, all degees ae equal and theefoe we obtain R(F k,p,s = 1. Assume that we ae given a sequence of integes = (n with = ω(n 1/. Let us pick a pime p with p n 4 and an odd intege k. Then we can conside the family F n k,p,s with any 0 s p/. Its vetex has size k(p + p + 1 = n n and by the monotonicity of /n and = ω(n 1/ we obtain that the sets of F k,p,s k+1 have size (p s =. 3 Concluding emaks As we mentioned in the Intoduction, the bound of Theoem 1.1 (i cannot be stengthened in geneal as the following example shows. If n =, then a maximal intesecting family F contains one set fom evey pai of complement sets. Thus the family F = F ( : 1 / F, F [ + 1, n]} []} is maximal intesecting and R(F = Θ( ( n = e Θ(n holds while = e Θ(n log n. 5

6 In Theoem 1. (iii, we could show egula maximal intesecting families only fo special values of n and. Thee ae two ways to genealize ou constuction. Fist, one needs not insist that all pojective planes should be of the same ode, but fo the maximality one still needs that they should be of the same asymptotic ode (one will have to choose s a bit moe caefully. This will uin the egulaity, but fo families F obtained this way R(F = 1 + o(1 would still hold. The othe possibility is to add exta vetices that do not belong to P i, similaly to the constuction used fo Theoem 1. (iii. This will enable us to obtain constuctions fo abitay values of n and (povided n is lage enough but fo these families F we will have R(F = Θ( n. It emains open whethe one can constuct maximal intesecting families with R-value 1 + o(1 fo any (n. Acknowledgement. This eseach was stated at the 3 d Emléktábla Wokshop held in Balatonalmádi, June 7-30, 011. Refeences [1] A. Blokhuis, On the size of a blocking set in P G(, p, Combinatoica, 14 (1994, [] I. Dinu, E. Fiedgut, Intesecting families ae essentially contained in Juntas, Combin. Pobab. Comput. 18 (009, [3] P. Edős, C. Ko, R. Rado, Intesection theoems fo systems of finite sets, Quat. J. Math. Oxfod, 1 (1961, [4] N. Lemons, C. Palme, The unbalance of set systems, Gaphs and Combinatoics 4 (008, [5] J. Nagua, On the inteval containing at least one pime numbe, Poceedings of the Japan Academy, Seies A 8 (195, [6] T. Szőnyi, Blocking Sets in Desaguesian Affine and Pojective Planes, Finite Fields and Thei Applications 3 (1997,

On the ratio of maximum and minimum degree in maximal intersecting families

On the ratio of maximum and minimum degree in maximal intersecting families On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting