The de Bruijn-Erdős Theorem for Hypergraphs

Transcription

1 The de Buijn-Edős Theoem fo Hypegaphs Noga Alon Keith E. Mellinge Dhuv Mubayi Jacques Vestaëte July 23, 2010 Abstact Fix integes n 2. A clique patition of ( is a collection of pope subsets A 1, A 2,..., A t such that ( Ai ( i is a patition of. Let cp(n, denote the minimum size of a clique patition of (. A classical theoem of de Buijn and Edős states that cp(n, 2 = n. In this pape we study cp(n,, and show in geneal that fo each fixed 3, cp(n, (1 + o(1n /2 as n. We conjectue cp(n, = (1 + o(1n /2. This conjectue has aleady been veified (in a vey stong sense fo = 3 by Hatman-Mullin-Stinson. We give futhe evidence of this conjectue by constucting, fo each 4, a family of (1 + o(1n /2 subsets of with the following popety: no two -sets of ae coveed moe than once and all but o(n of the -sets of ae coveed. We also give an absolute lowe bound cp(n, ( ( n / q+ 1 when n = q 2 + q + 1, and fo each chaacteize the finitely many configuations achieving equality with the lowe bound. Finally we note the connection of cp(n, to extemal gaph theoy, and detemine some new asymptotically shap bounds fo the Zaankiewicz poblem. 1 Intoduction A classical theoem of de Buijn and Edős [8] states that the minimum numbe of pope complete subgaphs (hencefoth cliques of the complete gaph K n that ae needed to patition its edge set is n. Equality holds only fo n 1 copies of K 2 on a vetex togethe with a clique of ode n 1 (a nea-pencil o the cliques whose vetex sets ae the q 2 + q + 1 point sets of lines in a pojective plane of ode q when n = q 2 + q + 1 and a pojective plane exists. Schools of Mathematics and Compute Science, Sackle Faculty of Exact Sciences, Tel Aviv Univesity, Tel Aviv 69978, Isael. nogaa@tau.ac.il Reseach suppoted in pat by an ERC Advanced gant and by a USA-Isaeli BSF gant Depatment of Mathematics, Univesity of May Washington, Fedeicksbug, VA kmelling@umw.edu Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois, Chicago, IL Reseach suppoted in pat by NSF gants DMS and DMS mubayi@math.uic.edu Depatment of Mathematics, Univesity of Califonia, San Diego, La Jolla CA jacques@ucsd.edu Reseach suppoted in pat by NSF gant DMS and a Hellman Fellowship 1

2 In this pape, we study the analog of the de Buijn-Edős theoem fo hypegaphs. Wite ( X fo the collection of -elements subsets of a set X; this is a clique. Thoughout this pape, we wite := {1, 2,..., n}. We will associate a hypegaph with its edge set. Definition. Fix integes n 2. A clique patition of a hypegaph H is a collection of pope subcliques that patition H. Let cp(n, be the minimum size of a clique patition of (. In othe wods, cp(n, is the minimum t fo which thee ae pope subsets A 1, A 2,..., A t such that ( Ai ( i is a patition of. The de Buijn-Edős Theoem can now be estated as cp(n, 2 = n togethe with the chaacteization of equality. In this pape, we conside cp(n, fo > 2. A minimum clique patition of shall be efeed to as an optimal clique patition. ( As noted above, thee ae essentially two types of configuations achieving equality in the de Buijn-Edős theoem: nea pencils and pojective planes. Fo = 3 and n = q thee is only one type of configuation that achieves equality, and this is called an invesive plane. An invesive plane is a pai (V, C whee V is a set of points and C is a set of subsets of points called cicles satisfying the following axioms: (1 any thee points of V must lie in exactly one cicle in C C, (2 fou points in V must exist that ae not contained in a common cicle and (3 fo any cicle C C and points p C and q V \C, thee is exactly one cicle D C such that p, q D and C D = {p}. It is known that if an invesive plane on n points exists, then n is necessaily of the fom q fo some intege q 2, and the numbe q is called the ode of the invesive plane. If the cliques in a clique patition fom an invesive plane, then we identify the patition with the invesive plane. The poblem of detemining cp(n, 3 is quite well undestood due to the following theoem. Theoem 1 (Hatman-Mullin-Stinson [17] Let q 3 be an intege and n = q Then If q is a pime powe, then cp(n, 3 = qn = n n 1, and if P is a clique patition of ( 3 Consequently, as n cp(n, 3 n n 1., P = qn, then P is an invesive plane of ode q. cp(n, 3 n 3/2. In Section 2, we shall give a simple desciption of invesive planes equiing only basic field aithmetic. Thee, we povide what might be consideed a classical model fo a finite invesive 2

3 plane, stating with the coodinates fo an affine plane ove a finite field. The model is not new (see, fo instance the vey complete book by Benz [5], o the moe ecent suvey by Wilke [29]. Although this teatment is not new, it appeas that it is not well-known. We include it hee fo completeness. In [17], cp(n, 3 is detemined when n is close to q 2 +1 and q is a pime powe, due to the existence of invesive planes and classification [26] of linea spaces with n points and m blocks such that (m n 2 n. In geneal, howeve, it is likely to be challenging to detemine cp(n, 3 exactly fo all n, in contast to the de Buijn-Edős Theoem. 1.1 Clique patitions of ( The poblem of detemining cp(n, fo > 3 appeas to be difficult. Fist we pesent a lowe bound fo cp(n, in tems of cp(n 1, 1: Theoem 2 Let > 2 and n > be integes. If cp(n 1, 1 = δcp(n, then ( ( δn n cp(n,. (1 Consequently, fo each fixed 2, and n, cp(n, (1 o(1n /2. Via constuctions we shall give evidence that this lowe bound may be the tue asymptotic behavio of cp(n, fo evey fixed 2. Theoem 3 Fo evey fixed 2 thee is a family H ( of -element sets such that H = o(n and H has a clique patition with (1 + o(1n /2 cliques. ( n It appeas to be difficult to extend the constuctions fo Theoem 3 to full patitions of ( without adding many moe subcliques. We nevetheless conjectue that in geneal, the above lowe bound is asymptotically shap: Conjectue 4 Fo evey fixed 2 we have cp(n, = (1 + o(1n /2 as n. 1.2 Chaacteization of clique patitions Equality holds in the de Buijn-Edős Theoem fo pojective planes and nea-pencils. The natue of clique patition numbes cp(n, fo > 2 suely depends on numbe theoetic popeties of n, so unlike in the de Buijn-Edős Theoem, a chaacteization of optimal patitions fo all n is likely to be moe difficult. Towads this goal, we pove the following theoem, which consides values of n that ae paameteized in a way that includes the de Buijn-Edős Theoem as a special case. As is customay, 3

4 we define the binomial coefficient ( x = x(x 1 (x + 1/! fo any positive intege and eal numbe x. Also, ecall that a Steine (n, k, t-system is a collection of k-element sets of such that evey t-element subset of lies in pecisely one of the k-element sets. Theoem 4 Let 2 and let n be a positive intege. Define the positive eal numbe q by the equation n = q 2 + q + 1. Then ( n cp(n, Equality holds if and only if one of the following holds: ( q+ 1. (2 n = + 1 and the patition is ( [+1] n > + 1 and = 2 and the patition is a pojective plane of ode q o a nea pencil the patition is a Steine (n, k, -system whee (n, k, {(8, 4, 3, (22, 6, 3, (23, 7, 4, (24, 8, 5}. Note that in the thid case above such Steine systems ae known to exist fo {3, 4, 5}, and so fo those values of we have a complete chaacteization of equality in Theoem 4. The case = 2 is the de Buijn-Edős Theoem. We note that not a single constuction of a Steine (n, m, -system is known fo any n > m > > 5. This is consideed to be one of the majo open poblems in design theoy [28]. 1.3 Zaankiewicz poblem Thee is a tight connection between cp(n, and the Zaankiewicz poblem fom extemal gaph theoy. The Zaankiewicz numbe z(m, n, s, t denotes the maximum possible numbe of 1s in an m n matix containing no s t mino consisting entiely of 1s. This can be ephased in tems of the maximum numbe of edges in an m n bipatite gaph containing no complete bipatite subgaph with s vetices in the pat of size m and t vetices in the pat of size n. Ou esults fo the clique patition numbe cp(n, imply the following new asymptotically shap esults fo Zaankiewicz numbes. Theoem 5 Fix 3. If m = (1 + o(1n /2, then z(m, n, 2, = (1 + o(1 n m as n. The special case = 3 had ealie been shown by Alon-Rónyai-Szabó [2], as pat of a moe geneal esult motivated by a question in discepancy theoy posed by Matouśek. Howeve, ou constuctions hee ae diffeent. Theoem 5 is in contast to a esult of Füedi [14] which shows that fo each fixed 2, we have z(m, n, 2, = (1 + o(1 ( 1n m wheneve m = (1 + o(1n. The poblem of detemining the asymptotic behavio of cp(n, fo fixed 2 seems moe challenging than that of detemining the Zaankiewicz numbes z(m, n, 2, fo m = (1 + o(1n /2. 4

5 1.4 Oganization In Section 2, we discuss invesive planes and give a self-contained pesentation of invesive planes. In Section 3, we discuss cp(n, fo > 3, stating in Section 3.1 with the poof of Theoem 2, poceeding in Section 3.2 with the poof of Theoem 3, and mentioning an altenative constuction fo {4, 5} in Section 3.3. In Section 4, we pove Theoem 4, and in Section 5 we point out the connection to the Zaankiewicz poblem and pove Theoem 5. 2 Invesive planes In this section, we give an elementay constuction of invesive planes of pime powe ode q 3 (mod 4 which may be of independent inteest. As mentioned in the intoduction, this pesentation is not new, but it appeas not to be well-known. Recall that the blocks of an invesive plane π ae called cicles and that if π has n points, then n is necessaily of the fom q fo some intege q 2, and the numbe q is called the ode of π. In an invesive plane of ode q, it is well-known that evey cicle has q + 1 points, evey point is in q(q + 1 cicles, and the total numbe of cicles is q 3 + q (see [10] fo moe infomation on invesive planes. The issue of constucting invesive planes of ode q has quite a long histoy beginning in the 1930s [30, 3, 4, 9] and it is known that invesive planes of all pime powe odes exist. We will now pesent an elementay pesentation of invesive planes. Constuction. Fo a pime powe q 3 (mod 4 and n = q 2 + 1, we conside the vetex set as (F q F q {v} := F 2 q {v}. Fo a = (a 1, a 2 F 2 q and λ F q \ {0}, define the cicle with cente a and finite adius λ to be C(a, λ = {(x 1, x 2 F 2 q : (x 1 a (x 2 a 2 2 = λ}. Thee ae q 3 q 2 such cicles, since thee ae q 2 choices fo a F 2 q and then q 1 choices fo λ. Fo each a F 2 q and µ F q define the following sets: C(a = {(x 1, x 2 F 2 q : x 2 a 2 = a 1 x 1 } {v} C(µ = {(x 1, x 2 F 2 q : x 1 = µ} {v}. It is convenient to efe to these sets as cicles too. Note that each of them has q + 1 points. These q 2 + q special cicles ae in one-to-one coespondence with the affine lines of F 2 q; just add {v}, a point at infinity, to each of the affine lines. The cicles C(a come fom lines with finite slope, wheeas the cicles C(µ come fom lines with infinite slope. The total numbe of sets defined in ou constuction is q 3 + q. It emains to show that all the cicles togethe fom an invesive plane of ode q, by veifying the thee axioms. In the Euclidean plane, evey thee points detemine a unique cicle unless they ae collinea, and the basis fo ou constuction is that this emains tue in finite fields. We make this pecise in the next esult. 5

6 Lemma 6 Evey thee non-collinea points in F 2 q lie in a unique cicle C(a, λ. No thee collinea points lie on a cicle C(a, λ. Poof: Let x = (x 1, x 2, y = (y 1, y 2, z = (z 1, z 2 be distinct non-collinea points in F 2 q. We will show that thee is a unique cicle C(a, λ that contains all thee of them with λ F q \ {0}. We wish to detemine the numbe of solutions (a, λ whee a F 2 q and λ F q \ {0} to the equations (x 1 a (x 2 a 2 2 = λ (3 (y 1 a (y 2 a 2 2 = λ (4 (z 1 a (z 2 a 2 2 = λ (5 Now (3 (4 and (4 (5 give ( 2(x1 y 1 2(x 2 y 2 2(y 1 z 1 2(y 2 z 2 ( a1 a 2 = ( x 2 1 y x2 2 y2 2 y 2 1 z2 1 + y2 2 z2 2. Since x, y, z ae non-collinea, the coefficient matix above is invetible, and hence thee is a unique a = (a 1, a 2 that satisfies the matix system above. Fo this paticula choice of a, define λ using (3. It is staightfowad to see that λ automatically satisfies both (4 and (5 and theefoe x, y and z detemine a unique cicle. It emains to show that λ 0. If λ = 0, then since q 3 (mod 4 we may apply Eule s Theoem which says that -1 is not a squae modulo q. Consequently, the only solution to equations (3, (4 and (5 is x = y = (a 1, a 2. This contadicts the fact that x, y, z ae distinct and theefoe λ 0. Now we show that if C(a, λ is a cicle then C(a, λ C(b 2 fo b = (b 1, b 2 F 2 q and also C(a, λ C(µ 2 fo µ F q. In the fist instance, (x 1, x 2 C(a, λ means that (x 1 a (x 2 a 2 2 = λ. Substituting x 2 = b 1 x 1 + b 2 above gives the quadatic equation (1 + b 2 1x c 1 x 1 + c 2 = 0 fo some c 1, c 2 F q. Since q 3 (mod 4, Eule s Theoem implies that 1 + b and hence the quadatic above has at most two solutions. Each of these solutions gives a unique solution fo x 2 and consequently C(a, λ C(b 2 as equied. The poof that C(a, λ C(µ 2 is simila. The above lemma gives a family P of q 3 + q sets in ( 3 when n = q with the popety that evey set of thee distinct points in is coveed by exactly one set. Indeed, the Lemma clealy shows this fo any thee points in F 2 q, since if they ae non-collinea they lie in a unique cicle C(a, λ, and if they ae collinea they lie in pecisely one affine line C(a \ {v} o C(µ \ {v}. If the thee points ae of the fom {x, y, v} with x, y F 2 q, then since {x, y} lies in a unique affine line, {x, y, v} lies in the unique extension of this line that has the fom C(a o C(µ. We now show that P is an invesive plane. By constuction, axioms 1 and 2 ae satisfied. To show that axiom 3 is satisfied we need an elementay counting agument using the fact that evey cicle has size q + 1. This will be poved via the following lemma which shows that each cicle C(a, λ has q + 1 points: 6

7 Lemma 7 Let q 3 mod 4 be a pime powe, and let (a, b F 2 q, λ F q \ {0}. Then the numbe of solutions (x, y F 2 q to the equation (x a 2 + (y b 2 = λ is exactly q + 1. Poof: Let S q = {(x, y F 2 q : (x a 2 + (y b 2 = λ}. By tanslation and scaling, S q = {(x, y F 2 q : x 2 + y 2 = δ} whee δ = χ(λ { 1, 1} and χ is the quadatic chaacte of F q. Theefoe S q = (1 + χ(x(1 + χ(y x+y=δ = q + x F q χ(x(δ x = q + x F q\{δ} = q + = q + x F q\{δ} w F q\{ 1} ( x χ δ x ( χ 1 + δ δ x χ(w = q χ( 1. Hee we used that 1 + δ/(δ x is a pemutation of F q \{ 1} and that w F q χ(w = 0. Since q 3 mod 4, χ( 1 = 1 and so S q = q + 1 We now show that axiom 3 is satisfied. So let C(a, λ be a cicle, u C(a, λ and v C(a, λ. Fo each point x C(a, λ \ {u} we have a cicle C x that contains u, v, x. Moeove, C x C x = {u, v} fo x x. Since these q cicles ae disjoint outside {u, v} and they all have size q + 1, thei union has size 2 + (q 1q = q 2 q + 2 = n (q 1. Theefoe we can find a point z outside the union of these cicles. But thee is a cicle C that contains u, v, z and so C C(a, λ = {u}. Moeove, C contains all the emaining q 1 points above, so it is the unique cicle that contains u, v and intesects C(a, λ at {u}. This poves axiom 3. 3 Clique patitions of ( In this section, we pove Theoems 2 and Poof of Theoem 2 Let P be an optimal clique patition of ( which has vetex set. Then P v = {C \ {v} : v C P } is a clique patition of ( [n 1] 1 so P v cp(n 1, 1. Noting the identity C = Pv, C P v 7

8 we have C n cp(n 1, 1. (6 C P On the othe hand since P is a clique patition of (, ( C C P = ( n. (7 By convexity of binomial coefficients, the ight hand side is a minimum when the C all equal thei aveage value, which by (6 is at least ncp(n 1, 1/cp(n, = δn by definition of δ. Inseting this in (7 gives ( Poof of Theoem 3 By the known esults about the distibution of pimes it suffices to pove the esult fo n = q 2, whee q is a pime powe. Fo othe values of n we can then take the constuction fo the smallest n > n satisfying n = q 2, with q being a pime powe, and conside the induced constuctions on n vetices among those n. Suppose, thus, that n = q 2, and let F = GF (q denote the finite field of size q. Let V be the set of all odeed pais (x, y with x, y F. Fo evey polynomial p(x of degee at most 1 ove F, let C p denote the subset C p = {(x, p(x : x F } of V. The collection C of all sets C p is a collection of q = n /2 subsets of V, each of size q = n. Clealy, no two membes of C shae moe than 1 elements, as two distinct polynomials of degee at most 1 can shae at most 1 points. Moeove, evey set {(x i, y i : 1 i } of points of V in which the x i -s ae paiwise distinct is contained in a unique set C p, as thee is a unique polynomial p of degee at most 1 satisfying p(x i = y i fo all i. Let H be the set of all -subsets of V contained in a membe of C. Then H has a clique patition with the n /2 cliques C p, and the only -sets in V that do not belong to H ae those that have at least two points (x, y with the same fist coodinate. The numbe of these -sets is at most ( ( q q 1 (1 + o(1q q 2 1 = (1 + o( ( 2! q2 1 = O(n 1/2 = o(n, completing the poof. 3.3 An altenative constuction fo {4, 5} In this subsection we pesent an altenative constuction that poves the assetion of Theoem 3 fo {4, 5}. Although this is less geneal than the pevious constuction we believe it is inteesting and may povide some exta insight. We wok in the classical finite pojective plane of ode q, denoted by π = P G(2, q. The plane π contains q 2 + q + 1 points that can be epesented by homogeneous coodinates (x, y, z. A 8

9 non-degeneate conic of π is a collection of points whose homogeneous coodinates satisfy some non-degeneate quadatic fom, and it is well-known that thee is exactly one such conic in π, up to isomophism. A typical example is the set of points satisfying the fom y 2 = xz which contains the points {(0, 0, 1} {(1, x, x 2 : x F q }. An ac is a set of points, no thee collinea, and it is staightfowad to show that conics fom acs. Moeove, a classical esult of Sege [23] says that when q is odd, evey set of q + 1 points, no thee collinea, is in fact a conic. Fo an oveview of esults on conics and acs in geneal, the eade is efeed to Chaptes 7 and 8 of [18]. It is well-known that thee ae pecisely q 5 q 2 conics in P G(2, q and that five points in geneal position (that is, no thee of which ae collinea detemine a unique conic. We can use these facts to cove 4-sets and 5-sets of π. The case = 4. Distinguish a special point P of π and conside the set of all conics passing though P. Ou points will be the points of π \ {P }. Hence, n = q 2 + q. Let X be the numbe of conics passing though an abitay point Q of π. By the tansitive popeties of Aut(π, it follows that X is independent of Q. We count pais (Q, C with Q a point of the conic C, by fist counting Q, and then counting C. This give us (q 2 + q + 1X = (q 5 q 2 (q + 1 = q 2 (q 1(q 2 + q + 1(q + 1. If follows that thee ae X = q 4 q 2 conics though a point of π. The numbe of 4-sets in the set π \ {P } is ( q 2 +q 4 = ( o(1q8. The numbe of 4-sets of π \ {P } coveed by the conics though P is (q 4 q 2 ( q 4 = ( o(1q8 (the diffeence is asymptotic to q7 which is of a lowe ode as q. Hence, we have shown that fo n = q 2 + q, thee is a collection of (1 o(1q 4 = (1 o(1n 2 pope edge-disjoint subcliques of ( 4 that cove all but o(q 8 = o(n 4 edges of ( 4. The case = 5. We can epeat ou constuction, using all points and conics of π, to obtain a simila esult fo = 5. The constuction hee is in fact simple than the one fo = 4. The total numbe of 5-sets of points in π is ( q 2 +q+1 5 = ( o(1q10. The numbe of 5-sets coveed by conics is (q 5 q 2 ( q+1 5 = ( o(1q10 (the diffeence is asymptotically 1 12 q9 which is of lowe ode. Hence, we have shown that fo n = q 2 + q + 1, thee is a collection of (1 o(1q 5 = (1 o(1n 5/2 pope edge-disjoint subcliques of ( 5 that cove all but o(q 10 = o(n 5 edges of ( 5. This completes the (second poof of Theoem 3 fo {4, 5}. 4 Poof of Theoem 4 We fist addess the lowe bound in Theoem 4. Befoe beginning the poof we need the following simple lemma, known as Chebyshev Sum Inequality (c.f., e.g., [16]. Lemma 8 (Chebyshev Sum Inequality Suppose that p > 1 is an intege and f, g : [p] Z + ae non-deceasing functions. Then p f(ig(i 1 p p f(i g(i. p 9

10 Thee ae seveal simple poofs of this inequality. A shot one is to obseve that the ight hand side is the expectation of the sum p f(ig(σ(i, whee σ is a andom unifomly chosen pemutation of {1, 2,..., p}, and that the maximum value of this sum is obtained when σ is the identity, since if σ(i > σ(j fo some i < j, then the pemutation σ obtained fom σ by swapping the values of σ(i and σ(j satisfies p f(ig(σ (i p f(ig(σ(i. 4.1 Poof of the lowe bound In this section we show the lowe bound in Theoem 4. Define (n := n(n 1... (n + 1 and φ(n, = ( ( n / q+ 1 (. Let P be a clique patition of whee n = q 2 + q + 1 and q is a positive eal numbe. We aim to show P φ(n,, and we poceed by induction on. Fo = 2, this follows fom the de Buijn-Edős Theoem since φ(n, 2 = q 2 + q + 1 = n. Since φ( + 1, = + 1, we may assume n > + 1. Suppose > 2 and let C 1, C 2,..., C p be the cliques in P, of espective sizes n 1 n 2 n p. Note as in (6, p n i nφ(n 1, 1. (8 Fo a set S, let P S = {C \ S : C P }. Then p ( ni = P 2 S. S ( 2 By the de Buijn-Edős Theoem, since PS is a clique patition of ( [n +2] 2, P S n + 2 and theefoe p ( ( ni n (n + 2 (9 2 2 with equality if and only if thee is a pojective plane on n +2 points, q is an intege, and evey clique in P has size q + 1. Note hee that we ae uling out the possibility of a nea pencil in evey PS, othewise P would have a clique of size n 1 and then P 1 + ( n 1 1 > φ(n, since n > + 1. On the othe hand since P is a clique patition of (, p ( ni = ( n. (10 Since n i fo all i, we may apply Lemma 8 with f(i = (n i 2 and g(i = (n i +2(n i +1 to obtain p (n i 1 p p (n i 2 (n i + 2(n i + 1. p By (9 and (10, we have p 1 n + 1 p (n i + 2(n i

11 The function h(x = (x + 2(x + 1 is convex fo x. Hence Jensen s inequality yields p (n i + 2(n i + 1 p(c + 2(c + 1 whee c is the aveage size of a clique in P. Consequently, n + 1 (c + 2(c + 1 and since n = q 2 + q + 1, we find c q + 1. Togethe with (8 we get p(q + 1 nφ(n 1, 1. Using the identity φ(n, (q + 1 = nφ(n 1, 1, this gives p φ(n, and completes the poof of the lowe bound in Theoem Equality in Theoem 4 Fo the constuctions giving equality in Theoem 4, we note that fo = 2 this is exactly the de Buijn-Edős Theoem. Moving on to 3, we have n = q 2 + q + 1, and equality holds in the above aguments if and only if equality holds in (9, which means that evey ( 2-set S has the popety that PS is an optimal patition ( \S 2. Theefoe evey -set is coveed at most (in fact exactly once and P is a Steine (n, q + 1, -system. It is well known (see [28] that the following divisibility equiements ae necessay fo the existence of such Steine systems: ( q + 1 i ( n i fo i = 0, 1,..., 1. (11 i i Equivalently, since n = q 2 + q + 1, (11 is equivalent to: i (q + i j j=1 i (q 2 + q + i j fo i = 0, 1,..., 1. (12 j=1 Fo each, let Q be the set of values of q pemitted by (12. Then the Q fom a deceasing chain Q 3 Q 4... and 1 Q fo all 3. Now Q 3 is the set of q such that q (q 2 + q q(q + 1 (q 2 + q + 1(q 2 + q q(q + 1(q + 2 (q 2 + q + 2(q 2 + q + 1(q 2 + q. The fist two conditions ae tivially satisfied, and the last is (q + 2 (q 2 + q + 2(q 2 + q + 1. This is equivalent to (q It follows that q {1, 2, 4, 10}, Q 3 = {1, 2, 4, 10} and then n {4, 8, 22, 112}. Since Steine (n, q + 2, 3-systems exist fo (n, q {(4, 1, (8, 2, (22, 4} (see van Lint and Wilson [28], Theoem 2 is tight in those cases and cp(4, 3 = 4 cp(8, 3 = 14 cp(22, 3 =

12 The case q = 10 may be uled out, since it is accepted (see [21] that a pojective plane of ode ten does not exist, hence it is impossible that PS is a patition of ( \S 2 fo S = 1. Fo = 4, we have the same divisibility equiements as = 3 togethe with q(q + 1(q + 2(q + 3 (q 2 + q + 3(q 2 + q + 2(q 2 + q + 1(q 2 + q. Equivalently, we ae looking fo q Q 3 satisfying (q + 2(q + 3 (q 2 + q + 3(q 2 + q + 2(q 2 + q + 1. This is not satisfied fo q {2, 10} and hence the only values q Q 3 = {1, 2, 4, 10} which satisfy these equiements ae q {1, 4} and Q 4 = {1, 4}. Since a Steine (23, 7, 4-system exists which coesponds to q = 4, we have cp(5, 4 = 5 cp(23, 4 = 253. Fo = 5, we ae seeking Steine (q 2 + q + 4, q + 4, 5-systems, which implies q {1, 4} = Q 5 and n {6, 24}. Since a Steine (24, 8, 5-system exists, we have the complete solution cp(6, 5 = 6 cp(24, 5 = 759. Finally we show Q = {1} fo 6 to complete the poof. Fist we have Q 5 = {1, 4} as just seen, and since Q Q 5 fo all 5, we only have to show Q 6 = {1}. If 4 Q 6, then the divisibility equiement (12 with i = 0 and q = 4 is This is false, since 3 is a pime facto with multiplicity thee on the left, and only two on the ight. Theefoe Q = {1} fo 6, and the only possible clique patition achieving equality in Theoem 4 has q = 1 and n = + 1, and theefoe it must be ( [+1]. 5 Zaankiewicz Poblem and Theoem 5 In this section we point out the connection between Zaankiewicz numbes and clique patitions. As a bypoduct, we pove Theoem 5. Recall that the Zaankiewicz numbe z(m, n, s, t is the maximum numbe of edges in an m n bipatite gaph containing no complete bipatite subgaph with s vetices in the pat of size m and t vetices in the pat of size n. The clique patition numbe cp(n, is elated to the Zaankiewicz numbes in the following sense: Lemma 9 Let n be a positive intege and m = ( ( n / k (. Then thee is a patition of into m cliques of size k if and only if z(m, n, 2, = km. Poof: We follow the classical agument of [20]. Fo any m n bipatite gaph with pats A and B, not containing a 2 complete bipatite subgaph (with the two vetex set lying in A, ( ( d(a B. (13 a A 12

13 By convexity, if e is the numbe of edges in the gaph we obtain ( ( e/m n m. (14 Suppose e = z(m, n, 2, and e > km. Then the above fomula becomes ( ( k n m <. Howeve this contadicts the identity elating m, n and k in the lemma. It follows that z(m, n, 2, km. Now conside the incidence gaph of points and sets of size k in the clique patition of (. This is an m n bipatite gaph, and since we have a clique patition, it does not contain a 2 complete bipatite subgaph. Theefoe z(m, n, 2, km. If z(m, n, 2, = km, then equality holds in (13 and (14, which means that d(a = k fo evey a A and evey -set of B is coveed exactly once. This gives us a patition of ( into m cliques of size k. The poof of Lemma 9 can be easily ewitten to pove the following: Fix 2 and let H ( satisfy H ( n o(n. Suppose that thee is a patition of H into m = (1 + o(1 ( ( n / k cliques of size k. Then z(m, n, 2, = (1 + o(1km. Using the peceding esults and constuctions in Sections 2 and 4, we obtain the following theoem on Zaankiewicz numbes which poves Theoem 5. Theoem 10 Let n = q 2 +1 and m = qn whee q is a pime powe. Then z(m, n, 2, 3 = (q 2 +qn. Futhemoe, if 4, n is an intege and m = (1+o(1n /2, then z(m, n, 2, = (1+o(1 n m as n. Poof: In the fist case we apply Lemma 9 with k = q + 1 and = 3. Such a clique patition exists by the constuction of invesive planes. Then we see that m = qn = ( ( n 3 / q+1 3 and so we immediately get z(m, n, 2, 3 = km = (q + 1qn = (q 2 + qn. Fo the cases 4 we use the asymptotic vesion of Lemma 9 above and the esults in Section 3.2. If n = q 2 with q a pime powe, the desied esult follows fom the constuction in the poof of Theoem 3. Fo othe values of n we take the smallest pime powe q so that q 2 n, conside the constuction fo that which is a bipatite gaph with vetex classes A and B, and take the expected numbe of edges in an induced subgaph on andomly chosen sets of sizes n and m in A and B, espectively. 6 Concluding emaks Clique patitions and extensions of Fishe s Inequality 13

14 The de Buijn-Edős Theoem is an extension of a well known inequality of Fishe [12] that assets that any nontivial clique patition of ( 2 in which all cliques have the same cadinality contains at least n cliques. Fishe s Inequality has been extended in seveal ways. One such extension is due to Ray-Chaudhui and Wilson [22] who poved that fo even, any clique patition of ( in which all cliques have the same cadinality contains at least ( n /2 cliques. Ou bound hee is stonge (by a facto of (1 + o(1(/2! fo even, and applies to odd values of as well without having to assume that all cliques have the same cadinality. On the othe hand, the esult in [22], whose poof is algebaic, holds even if evey -set is coveed exactly λ times fo some λ > 0. Patitions into complete -patite -gaphs A well known esult of Gaham and Pollak [15] assets that the complete gaph on n vetices can be edge patitioned into n 1, but not less, complete bipatite gaphs. This can be viewed as a bipatite analogue of the de Buijn-Edős Theoem. In the bipatite case, unlike the one dealing with clique-patitions, thee ae many extemal configuations, and the only known poofs of the lowe bound ae algebaic (though ecently Vishwanathan [27] pesented a counting agument that eplaces the linea algebaic pat of one of these poofs. The -patite vesion of the poblem consideed hee is to detemine the minimum possible numbe of complete -patite -gaphs in a decomposition of the edges of the complete -gaph on n vetices. Moe fomally, this is the minimum p = p(n, so that thee ae p collections of the }, (1 i p, satisfying A (i A (i s = fo all 1 i p and all fom {A (i 1, A(i 2,..., A(i 1 j < s, A (i j fo all admissible i and j, and fo evey -subset R of n thee is a unique i, 1 i p so that R A (i j = 1 fo all 1 j. Thus, the Gaham Pollak esult assets that p(n, 2 = n 1. In [1] it is poved that p(n, 3 = n 2 and that fo evey fixed thee ae two positive constants c 1 (, c 2 ( so that c 1 (n /2 p(n, c 2 (n /2 fo all n (see also [6] fo slight impovements. Note that fo even the exponent of n is the same as the one appeaing in ou bounds fo the clique patition poblem (that is, the function cp(n, discussed hee, but fo odd values of, and in paticula fo = 3 whee both functions ae well undestood, the exponents diffe. Geometic desciption of invesive planes. j Invesive planes ae moe widely modeled using the techniques of finite pojective geomety. Fo completeness, we biefly descibe the known models hee since these planes povide optimal configuations. Invesive planes can be constucted fom ovoids. These ae sometimes called egg-like invesive planes (o Miquelian planes when the ovoid is an elliptic quadic as we descibe below. Thas [24] showed that fo odd q, if P is an optimal clique patition of ( 3 and n = q 2 +1, and fo some v the vetex sets V (C\{v} : C P constitute the point sets of the lines of the affine plane AG(2, q, then the vetex sets of the cliques in P ae the cicles of an egg-like invesive plane. It follows that the constuction given in Section 2.1 is equivalent to the known constuction of invesive planes when the field has odd chaacteistic q. To descibe the ovoidal constuction, the setting is finite pojective 3-space P G(3, q. An elliptic quadic of P G(3, q foms a set of q points, no thee of which ae collinea; in geneal, such a set of points in P G(3, q is called an ovoid. Now conside the points of an ovoid O of P G(3, q as the vetices of ( [q 2 +1] 3 togethe with the non-tangent plana coss sections of O. As thee points uniquely detemine a plane, each set of thee points of O (each edge of the hypegaph is 14

15 coveed by one of these plana coss sections. Moeove, it is staightfowad to show that each non-tangent plane meets O in q + 1 points, foming a plana conic, an object we descibed in Section 3.3. It follows that the set of points of O togethe with the plana coss-sections of O fom an invesive plane. It is an impotant open question in pojective geomety as to whethe fo any q, an invesive plane of ode q must be constucted fom an ovoid as above. Balotti [3] and Dembowski [9] showed that if q is odd, then any ovoid is pojectively equivalent to an elliptic quadic the set of points satisfying a quadatic fom xy + φ(w, z (hee φ is an ieducible quadatic fom ove F q. In this case, thee is an altenative desciption: conside the gound set to be F q 2 { } and conside the images of the set F q { } unde the goup of pemutations P GL(2, q 2 = { z azα + b cz α + d : ad bc 0, α Aut(F q 2 }. The q 3 + q images ae taken to be the cicles, each with q + 1 points, and it can be shown that the elliptic quadic constuction is equivalent to the semi-linea factional goup constuction. Note that this model can also be viewed as the points of P G(1, q 2 with blocks coesponding to the Bae sublines (i.e., copies of P G(1, q theein. Fo q even thee exist ovoids that ae not elliptic quadics, the so-called Tits ovoids [25]. Howeve, Dembowski s Theoem [9] shows that if q is even, then any invesive plane is deived fom an ovoid as above, and so in this sense, optimal clique patitions of ( 3 when n = q and q = 2 h fo h 2 ae unique up to the choice of the ovoid. It follows fom the peviously mentioned esult of Thas [24], tanslated into the language of clique patitions, that fo odd q, if P is an optimal clique patition of ( 3 and n = q 2 + 1, and some Pv is the classical affine plane AG(2, q, then P itself is constucted as above fom an elliptic quadic. In paticula, the optimal clique patitions ae unique. It appeas difficult, howeve, to claim that any of the clique patitions Pv ae isomophic to the classical affine plane. Much moe on finite pojective 3-space, including esults on ovoids and quadics, can be found in the book by Hischfeld [19]. Classification of linea spaces. Theoem 2 gives a geneal lowe bound on cp(n, in tems of cp(n 1, 1. We concentate on the case = 3. A necessay condition fo tightness in Theoem 4 fo = 3 when n = q 2 + q + 2 and q is a positive intege is the existence of a pojective plane of ode q on n 1 points. If n is an intege such that no pojective plane on n 1 points exists, this aises the question as to the minimum 2-designs which ae not pojective planes and not nea pencils on n 1 points. A complete analysis of 2-designs on v points with b blocks such that (b v 2 v was caied out by Totten [24] (1976 (see also [13]. In paticula, those 2-designs ae one of the following: 15

16 (a nea pencils (b an affine plane of ode q with a linea space on at most q + 1 new points at infinity (add a point to evey line in a paallel class, and then amongst the new points ceate a 2-design of lines this in paticula contains the pojective planes when q + 1 points ae added and a single line though them is added (c embeddable in a pojective plane (delete at most v + 1 points fom a pojective plane, deleting any line which becomes a singleton (d an exceptional configuation with v = 6 and b = 8. We point out that Theoem 2 can be tight. Suppose n = 21. By the classification of 2-designs, Pv is at least the numbe of lines in an affine plane of ode q = 4 togethe with the tivial linea space on q points at infinity, so cp(n 1, 2 q 2 + q + 1 i.e. cp(20, Using this in Theoem 2, if x = cp(21, 3 we obtain ( (21 2 ( /x 21 x 3 3 which gives x ( > 76. Since x is an intege, x 77. Howeve (see Theoem 4 cp(22, 3 = 77 due to the existence of the Steine (22, 6, 3-system, so we conclude x cp(22, 3 = 77. It follows that cp(21, 3 = 77. While it is possible to genealize the agument we just used fo n = 21 by using the classification of 2-designs, it is the lack of constuctive uppe bounds whee moe wok is needed, and in geneal thee is a gap between the uppe and lowe bounds fo cp(n, 3. An extensive suvey of the existence poblem fo Steine systems may be found in [7]. Refeences [1] N. Alon, Decomposition of the complete -gaph into complete -patite -gaphs, Gaphs and Combinatoics 2 (1986, [2] N. Alon, L. Rónyai, T. Szabó, Nom-gaphs: vaiations and applications. J. Combin. Theoy Se. B 76 (1999 no. 2, [3] A. Balotti, Un estensione del teoema di Sege-Kustaanheimo, Boll. U.M.I., 10 (1955, [4] W. Benz, Uebe Mobiusebenen. Ein Beicht, Jahesb. D.M.V., 63 ( [5] W. Benz, Volesungen be Geometie de Algeben, Geometien von Mbius, Laguee-Lie, Minkowski in einheitliche und gundlagengeometische Behandlung. Die Gundlehen de mathematischen Wissenschaften, Band 197. Spinge-Velag, Belin-New Yok (1973. [6] S. Cioaba, A. Kündgen, J. Vestaëte, On decompositions of complete hypegaphs, J.Combin. Theoy Seies A, 116 (2009, [7] C. Colboun, R. Mathon, Steine systems: in Handbook of Combinatoial Designs, second edition, (

17 [8] N. G. de Buijn, P. Edős, On a combinatoial poblem. Nedel. Akad. Wetensch., Poc. 51, ( = Indagationes Math. 10, ( [9] P. Dembowski, Invesive planes of even ode. Bull. Ame. Math. Soc. 69 ( [10] P. Dembowski, D. R. Hughes, On finite invesive planes. J. London Math. Soc. 40 ( [11] P. Edős, R. C. Mullin, V. T. Sós, D. Stinson: Finite linea spaces and pojective planes, Discete Math. 47 (1983 no. 1, [12] R. A. Fishe, An examination of the diffeent possible solutions of a poblem in incomplete blocks, Annals of Eugenics 10 (1940, [13] J. Fowle, A shot poof of Totten s classification of esticted linea spaces, Geometiae Dedicata, 15 (1984 no. 4, [14] Z. Füedi, New asymptotics fo bipatite Tuán numbes. J. Combin. Theoy Se. A 75 (1996 no. 1, [15] R.L. Gaham and H.O. Pollak, On the addessing poblem fo loop switching, Bell System Tech. 680 J. 50 (1971, [16] G. H. Hady, J. E. Littlewood and G. Polya, Inequalities, 2nd edition, Cambidge Univesity Pess (1988, Cambidge, England, pp [17] A. Hatman, R. C. Mullin, D. Stinson, Exact coveing configuations and Steine Systems, J. London. Math. Soc (2 25, ( [18] J. Hischfeld, Pojective geometies ove finite fields, Second edition. Oxfod Mathematical Monogaphs. The Claendon Pess, Oxfod Univesity Pess, New Yok (1998. [19] J. Hischfeld, Finite pojective spaces of thee dimensions, Oxfod Mathematical Monogaphs. The Claendon Pess, Oxfod Univesity Pess, New Yok (1985. [20] T. Kővái, V. T. Sós and P. Tuán, On a poblem of K. Zaankiewicz, Colloquium Math., 3, (1954, [21] C. Lam, C. The seach fo a finite pojective plane of ode 10. Ame. Math. Monthly 98 (1991, no. 4, [22] D. K. Ray-Chaudhui and R. M. Wilson, On t-designs, Osaka J. Math. 12 (1975, [23] B. Sege, Ovals in a finite pojective plane, Canad. J. Math. 7 ( [24] J. Thas, The affine plane AG(2,q, q odd, has a unique one point extension, Invent. math. 118 ( [25] J. Tits, Ovoïdes et goupes de Suzuki, Ach. Math. 13 ( [26] J. Totten, Classification of esticted linea spaces, Canad. J. Math. 28 (1976, [27] S. Vishwanathan, A counting poof of the Gaham-Pollak Theoem, pepint 17

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