Families with no matchings of size s

Transcription

1 Families wih no machings of size s Peer Franl Andrey Kupavsii Absrac Le 2, s 2 be posiive inegers. Le be an n-elemen se, n s. Subses of 2 are called families. If F ( ), hen i is called - uniform. Wha is he maximum size e (n, s) of a -uniform family wihou s pairwise disjoin members? The well-nown Erdős Maching Conjecure would provide he answer for all n,, s in he above range. For n > 2s i is nown ha he maximum is aained by A 1(T ) := {A : A =, A T } for some fixed (s 1)-elemen se T X. We discuss recen progress on his problem. In paricular, our recen sabiliy resul saes ha for n > (2 + o(1))s and a -uniform family F, F A 1(T ), hen F is considerably smaller. This resul is applied o obain he corresponding ani-ramsey numbers in a wide range. Removing he condiion of uniformness, we arrive a anoher classical problem of Erdős, which was solved by Kleiman for n 0 or 1 (mod s). We succeeded in resolving his long-sanding problem for n 2 (mod s) via a new averaging echnique which migh prove useful in various oher siuaions. 1 INTRODUCTION Pu := {1, 2,..., n} and le 2 denoe he power se of. A subse F 2 is called a family of subses of, or simply a family. For 0 n we use he noaion ( ) := {H : H = }. The maximum number of pairwise disjoin members of a family F is denoed by ν(f) and is called he maching number of F. Le us define he following wo quaniies for n,, s 2: e(n, s) := max { F : F 2, ν(f) < s }, { ( ) } e (n, s) := max F : F, ν(f) < s. Alfréd Rényi Insiue of Mahemaics, Budapes, Hungary; peer.franl@gmail.com Moscow Insiue of Physics and Technology, Ecole Polyechnique Fédérale de Lausanne; upavsii@yandex.ru Research suppored by by he gran RNF

2 For s = 2 boh quaniies were deermined by Erdős, Ko and Rado [4]: e(n, 2) = 2 n 1, e (n, 2) = ( n 1 1) for n 2. For s 3 his becomes a much harder as. Le us firs discuss he -uniform problem. 2 THE UNIFORM CASE. STABILITY The following families are he naural candidaes for being an exremal family of -ses wih no (s + 1)-maching: { ( ) A () i (n, s) := A : A [(s + 1)i 1] i }, 1 i. (1) Conjecure 1 (Erdős Maching Conjecure [2]). For n (s + 1) we have e (n, s + 1) = max { A () 1 (n, s), A() (n, s) }. (2) Conjecure 1 is nown o be rue for 3 (cf. [3], [12] and [6]). Improving some earlier resuls, in [5] i is shown ha ( ) ( ) e (n, s+1) = A () n n s 1 (n, s) = for n (2s+1) s. (3) Alhough we did no progress on he conjecure, we advanced in undersanding he srucure of families wih no s-machings ha have cardinaliies close o e (n, s). Le us define he covering number τ(f): τ(f) := min{ C : C, F F C F }. Noe ha τ(a () 1 (n, s)) = ν(a() 1 (n, s)) = s for n (s + 1). In he case s = 2 one has a very useful sabiliy heorem due o Hilon and Milner [10]. Below we discuss his heorem ogeher wih is naural generalizaion o he case s 2. Pu X := [s + 1, s + ] and consider he following family: { H () (n, s) := X H ( ) } : H [s 1] or s H, H X. Noe ha ν(h () (n, s)) = s < τ(h () (n, s)) for n s and ( ) ( ) ( ) n n s n s H () (n, s) = + 1. (4) 1 Theorem (Hilon-Milner [10]). Suppose ha n 2 and le F ( ) be a family saisfying ν(f) = 1 and τ(f) 2. Then F H () (n, 1) holds. The following generalizaion of he Hilon-Milner heorem for s 2 is obained in [8]. Theorem 1 (Franl, Kupavsii [8]). Suppose ha 3, n ( 2 + o(1) ) s, where o(1) 0 as s. Then for any G ( ) wih ν(g) = s < τ(g) we have G H () (n, s). 2

3 Informally, i saes ha any family of size larger han H () (n, s) mus be a subfamily of A () 1 (n, s). We menion ha for n 23 s his heorem was proven by Bollobás, Dayin and Erdős [1]. 3 AN APPLICATION TO AN ANTI-RAMSEY PROBLEM Theorem 1 seems o be useful o aac oher hypergraph problems. In [8] we apply a similar resul o he following ani-ramsey problem. Consider a coloring of ( ) wih M colors (each color mus be presen in he coloring). The ani-ramsey number ar(n,, s) is he minimum M such ha in any such coloring here is a rainbow s-maching, ha is, a se of s pairwise disjoin -ses from pairwise disinc color classes. This quaniy was sudied by Özahya and Young [13], who have made he following conjecure. Conjecure 2 ([13]). One has ar(n,, s) = e (n, s 1) + 2 for all n > s. I is no difficul o see ha ar(n,, s) e (n, s 1) + 2 for any n,, s. Indeed, consider he larges family of -ses wih no (s 1)-maching and assign a differen color o each of is ses. Nex, assign one new color o all he remaining ses. This is a coloring of ( ) in e (n, s 1) + 1 colors wihou a rainbow s-maching. In [13] he auhors proved his conjecure for s = 3 and for n 2 3 s. They also obained he bound ar(n,, s) e (n, s 1)+s for n s+(s 1)( 1). In [8] we prove he following heorem. Theorem 2 (Franl, Kupavsii [8]). We have ar(n,, s) = e (n, s 1) + 2 for n s + (s 1)( 1), 3. We remar ha he case = 2 of Conjecure 2 has already been seled for all values of parameers (see [13] for he hisory of he problem). Theorem 2 is acually a consequence of a much sronger resul proven in [8], we refer o [8] for deails. 4 THE NON-UNIFORM CASE Le us discuss he quaniy e(n, s) in his secion. Wha are he families in 2 wih no s-machings? One naural example of such family is ( m) for m = n+1 s. Erdős conjecured ha for n = sm 1 his family is exremal. Half a cenury ago Kleiman proved his conjecure and also deermined e(sm, s). Theorem (Kleiman [11]). e(sm 1, s) = m sm 1 ( ) sm 1 e(sm, s) = + m ( ) sm 1, (5) ( ) sm. (6) m+1 sm 3

4 For s = 2 boh formulae give 2 n 1, he easy-o-prove bound from he Erdős- Ko-Rado heorem. In he case s = 3 here is jus one case no covered by he Kleiman Theorem, namely n 1 ( mod 3). This was he subjec of he PhD disseraion of Quinn [14]. There he gave a very long and edious proof of he following equaliy: e(3m + 1, 3) = ( ) 3m m 1 + m+1 3m+1 ( 3m + 1 ). (7) Unforunaely, his resul was never published and no furher progress was made on he problem unil recenly. In [7] and [9] we deermined e(n, s) for a relaively wide range of parameers. For he sae of breviy, we sae he second par of he nex heorem somewha imprecisely. Theorem 3 (Franl, Kupavsii, [7], [9]). 1. Suppose ha n = sm+s 2. Then ( ) n 1 e(n, s) = + ( ) n. (8) m 1 m+1 n 2. Deerminaion of e(sm + s l, s) for all s lm + 3l + 3. To prove he firs par of Theorem 3, we used wo differen mehods for s 4 and s 5. Below we sech he idea of he proof in he case s = 3, which provides us wih a relaively shor proof of Quinn s resul [14]. We hin ha he proof mehod may be useful for oher problems ha as for he larges families in 2 wihou a cerain configuraion. Idea of he proof for n = 3m + 1, s = 3. The proof is by an averaging argumen. We sar wih a family F wih no 3-maching. We assume ha F is monoone: if i conains a se, hen i conains all is superses. Then we fix a permuaion σ and a family H(σ), which we call a es configuraion. The family H(σ) conains all ( m)-ses ha form inervals in σ. Moreover, for each riple of pairwise disjoin m-ses, we ae several (m+i)-ses, i = 1, 2, 3, which conain one of he m-ses and are disjoin wih pairs of some of he ( m)-ses, creaing new pairwise disjoin riples in H(σ). For a precise descripion of H(σ), see [9]. Is choice is essenial for he proof. One imporan feaure is ha H(σ) splis ino n subfamilies, each of which is very similar o a chain wih respec o conainmen, each passing hrough exacly one m-se. The nex sep is o analyze F H(σ). The family H(σ) is consruced in such a way ha F inersecs i in jus a righ amoun of ses. To formalize i, we inroduce charges and use a discharging mehod. We give charges o each se in F H(σ), and hen redisribue hem beween he ses in H(σ) in order o show ha he charge of F H(σ) is no oo large. In he redisribuion par we ransfer he weigh from he ( m)-ses o he ( m+1)-ses. Once compleed, we average over σ o show ha F canno be larger han he bound from he heorem. We pu differen charges on he ses in F H(σ) o compensae for he fac ha he proporion of i-ses conained in H(σ) is differen for differen i, which shows up when we do he averaging. 4

5 References [1] B. Bollobás, D.E. Dayin, P. Erdős, Ses of independen edges of a hypergraph, Quar. J. Mah. Oxford Ser. 27 (1976), N2, [2] P. Erdős, A problem on independen r-uples, Ann. Univ. Sci. Budapes. 8 (1965) [3] P. Erdős, T. Gallai, On maximal pahs and circuis of graphs, Aca Mah. Acad. Sci. Hungar. 10 (1959), [4] P. Erdős, C. Ko, R. Rado, Inersecion heorems for sysems of finie ses, The Quarerly Journal of Mahemaics, 12 (1961) N1, [5] P. Franl, Improved bounds for Erdős Maching Conjecure, Journ. of Comb. Theory Ser. A 120 (2013), [6] P. Franl, On he maximum number of edges in a hypergraph wih given maching number, arxiv: [7] P. Franl, A. Kupavsii, Families wih no s pairwise disjoin ses, o appear in he Journal of London Mahemaical Sociey, arxiv: [8] P. Franl, A. Kupavsii, Two problems of P. Erdős on machings in se families, submied, arxiv: [9] P. Franl, A. Kupavsii, The larges families of ses wih no maching of sizes 3 and 4, arxiv: [10] A.J.W. Hilon, E.C. Milner, Some inersecion heorems for sysems of finie ses, Quar. J. Mah. Oxford 18 (1967), [11] D.J. Kleiman, Maximal number of subses of a finie se no of which are pairwise disjoin, Journ. of Comb. Theory 5 (1968), [12] T. Lucza, K. Mieczowsa, On Erdős exremal problem on machings in hypergraphs, Journ. of Comb. Theory, Ser. A 124 (2014), [13] L. Özahya, M. Young, Ani-Ramsey number of machings in hypergraphs, Discree Mahemaics 313 (2013), N20, [14] F. Quinn, PhD Thesis, Massachuses Insiue of Technology (1986). 5