Caps and Colouring Steiner Triple Systems

Transcription

1 Desgns, Codes and Cryptography, 13, (1998) c 1998 Kluwer Academc Publshers, Boston. Manufactured n The Netherlands. Caps and Colourng Stener Trple Systems AIDEN BRUEN* Department of Mathematcs, Unversty of Western Ontaro, London, ON, N6A 5B7 Canada LUCIEN HADDAD Department of Mathematcs and CS, Royal Mltary College, P.O. Box 17000, STN Forces, Kngston, ON, K7K 7B4, Canada DAVID WEHLAU Department of Mathematcs and CS, Royal Mltary College, P.O. Box 17000, STN Forces, Kngston, ON, K7K 7B4, Canada Communcated by: J. W. P. Hrschfeld Receved October 3, 1994; Accepted October 25, 1996 Abstract. Hll [6] showed that the largest cap n PG(5, 3) has cardnalty 56. Usng ths cap t s easy to construct a cap of cardnalty 45 n AG(5, 3). Here we show that the sze of a cap n AG(5, 3) s bounded above by 48. We also gve an example of three dsjont 45-caps n AG(5, 3). Usng these two results we are able to prove that the Stener trple system AG(5, 3) s 6-chromatc, and so we exhbt the frst specfc example of a 6-chromatc Stener trple system. Keywords: caps, Stener trple systems, colourng Let F n 3 denote the vector space of dmenson n over F 3, the feld of order 3, and let AG(n, 3) be the set of all cosets of F n 3. Then AG(n, 3) s called the affne geometry of dmenson n over F 3.Fork=0,...,n,ak-flat of AG(n, 3) s a coset of a subspace of dmenson k. The ponts of AG(n, 3) are the 0-flats and they are dentfed wth the vectors of F n 3. The projectve geometry PG(n, 3) s defned as the space of equvalance classes (AG(n + 1, 3)\{ 0})/ where x y f c F 3 such that x = cy. Fork 0, the mage of a (k +1)-flat n AG(n, 3) s defned to be a k-flat of PG(n, 3). In both AG(n, 3) and PG(n, 3), the 1-flats are called lnes, the 2-flats are called planes and the (n 1)-flats are called hyperplanes. A subset of AG(n, 3) or PG(n, 3) s called a cap f no three of ts ponts are collnear,.e., f no three of ts ponts le n the same 1-flat. A cap of cardnalty k s called a k-cap. We wll need some results about the sze and structure of caps n AG(4, 3) and AG(5, 3). We denote by β(ag(n,3)) the largest nteger k for whch there exsts a k-cap n AG(n, 3). It s easy to see that β(ag(1,3)) = 2 and β(ag(2,3)) = 4. That β(ag(3,3)) = 9 s well known (see [8]). Pellegrno [9] showed that β(ag(4,3)) = 20. In ths paper we show that 45 β(ag(5,3)) 48. A Stener trple system of order v (an STS(v)) s a par S = (V (S), T ) where V (S) s a v-set and T s a collecton of trples (subsets of V (S) of cardnalty 3) such that each par of dstnct elements of S les n exactly one trple of T. It s well known that a STS(v) * Ths research s partally supported by NSERC Grants.

2 52 BRUEN, HADDAD AND WEHLAU exsts ff v 1 or 3 (mod 6), such an nteger s called admssble. Moreover t s easy to verfy that the set of all ponts of AG(n, 3) together wth ts lnes forms an STS(3 n ). An r-colourng of a STS, S = (V (S), T ), s a parttonng of the vertex set V (S) nto r dsjont subsets such that no trple of T s contaned entrely n one of these r subsets. If an STS(v), S, has an r-colourng but no (r 1)-colourng then we say that S s r-chromatc. We pont out n passng that the exstence of STS s wth prescrbed chromatc number r 5 s far from beng straghtforward and s shown (only) by non-constructve methods n [1], and moreover, the frst specfc example of a 5-chromatc STS s gven n [3]. It s often extremely hard to show that a gven STS cannot be k-coloured when k 4 (cf. the calculaton done n [3]). Here we wll gve a relatvely short proof that AG(5, 3) does not admt a 5-colourng. Let φ(n) denote the number of hyperplanes n AG(n, 3), and for j = 1, 2, let φ(j,n) denote the number of dstnct hyperplanes passng through j fxed ponts. It s well known (e.g., see [10]) that φ(2,n) = 3 n 3 3 n 1 3 n 2 3, φ(1,n) = 3 n 1 n 3 3n 2 3 n 1 3n 2, φ(n) = 3φ(1,n). Now f 3 ponts do not form a lne, then they determne a unque plane and there are (3 n 3 l ) 2 l n 2 (3 n 1 3 l ) dstnct hyperplanes through any plane n AG(n, 3) where n 4 (ths number s clearly one f n = 3). Let n 4 and C AG(n, 3) be a set of cardnalty t. For 0 t, defne n := {Ksuch that K s a hyperplane and K C =}. Consder the followng equatons (see [4] and [7]): Clearly n = 3 n 1 3 n 1 3 n 1 3 n 2 and countng the number of pars {p, K } where p s a pont of the hyperplane K and p C we fnd n = 3 n 2 3 n 1 3 n 1 3 n 2t. Now countng trples {p 1, p 2, K } such that K s a hyperplane and {p 1, p 2 } C Kgves ( ) n = 3 n 3 3 n ( ) 1 t. 2 3 n 2 3 n 3 2 Moreover suppose that the set C s a cap, then by countng quadruples {p 1, p 2, p 3, K } where K s a hyperplane and {p 1, p 2, p 3 } C Kwe obtan ( ) n = (3 n 3 l ( ) ) t. 3 (3 n 1 3 l ) 3 2 l n 2 Consder now a cubc polynomal ( P() := ( r 1 )( r 2 )( r 3 ). Then there are numbers a 3,...,a 0 such that P() = a ( 3 3) + a2 2) + a1 + a 0. Thus ( ) ( ) P()n = a 3 n + a 2 n + a 1 n +a 0 n 3 2

3 CAPS AND COLOURING STEINER TRIPLE SYSTEMS 53 = a 3 2 l n 2 (3 n 3 l ) (3 n 1 3 l ) ( ) t + a 2 3 n 3 3 n n 2 3 n 3 ( ) t 2 + a 1 3 n 2 3 n 1 3 n 1 3 t + a 03 n 1 3 n 1 n 2 3 n 1 3. n 2 Ths sum s a cubc polynomal of t, whch s denoted f P (t). We may use f P (t) to study C. For example we may choose the roots r 1, r 2 and r 3 of P() such that f n 0 then P() 0. Wth such a choce, f P (t) s necessarly nonnegatve and ths gves nformaton about the value of t. Now Pellegrno showed n [9] that the largest caps n PG(4, 3) have cardnalty 20. Snce one of the caps he constructed actually les n AG(4, 3) PG(4, 3), ths proved β(ag(4, 3)) = 20. In [5] Hll classfed all nonsomorphc 20-caps n PG(4, 3). Only one of the nequvalent 20-caps s contaned n an affne subspace of PG(4, 3), and ts ntersectons wth the 120 dfferent hyperplanes of AG(4, 3) gve the followng values for the n n 2 = 10, n 6 = 60, n 8 = 30, n 9 = 20 and n = 0 for n 2, 6, 8, 9. (1) A drect proof of these facts can be found n [7]. It follows easly from (1) that f C s a 20-cap n AG(4, 3), and L 1, L 2 and L 3 are three parallel hyperplanes, then { C L 1, C L 2, C L 3 } {{2, 9, 9}, {6, 6, 8}}. Moreover, n [6], Hll shows that the largest cap n PG(5, 3) has cardnalty 56 and ths cap s unque up to somorphsm. Ths fact s reported n the concluson of [2] where t s conjectured that β(ag(5,3)) = 45. We wll show that 45 β(ag(5,3)) 48. Hll showed that f C s hs 56-cap and H s any hyperplane of PG(5, 3) then C H {11, 20}. If we choose H wth C H =11 then C\(C H) s a 45-cap n (PG(5, 3)\H) = AG(5, 3). To prove that β(ag(5,3)) 48 we frst need the followng lemma. LEMMA 1 Let C be a 49-cap n AG(5, 3). Then there s a hyperplane H of AG(5, 3) such that C H =9. Proof. Frst note that there s no hyperplane H 0 such that C H 0 8, as otherwse one of the two hyperplanes parallel to H 0 would have to contan at least 21 elements of C. Consder now the polynomal P() := ( 11)( 17)( 18). Note that P() 0 for all ntegers 11. Thus n P 0 for all ntegers except possbly = 9, 10. Now P()n = f P (49) = 92. (2) Hence at least one of n 9 or n 10 s nonzero. Now f H 1 s any hyperplane such that C H 1 = 10, then the two hyperplanes H 2 and H 3 parallel to t satsfy { C H 2, C H 3 }={19, 20}. Thus n 19 n 10 and n 20 n 10. Suppose now that n 9 = 0. Then from (2) we have P(10)n 10 + P(19)n 19 + P(20)n 20 + A = 92, where A 0. As n 19 n 10 and n 20 n 10, P(10) = 56, P(19) = 16 and P(20) = 54, we deduce that 92 P(10)n 10 + P(19)n 19 + P(20)n 20 P(10)n 10 + P(19)n 10 + P(20)n 10 = ( )n 10,a contradcton. Hence n 9 0.

4 54 BRUEN, HADDAD AND WEHLAU LEMMA 2 β(ag(5,3)) 48 Proof. Let C be a cap n AG(5, 3) of sze 49. By the lemma above, there s a hyperplane H 1 such that C H 1 =9. Now the two hyperplanes H 2 and H 3 parallel to H 1 satsfy C H 2 = C H 3 =20, and thus C H s a maxmal cap of H for = 2, 3. Take K 1 another hyperplane wth K 2 and K 3 the two hyperplanes parallel to K 1, and defne L j := H K j = AG(3,3) for 1, j 3. In addton to H = L 1 L 2 L 3 and K j = L 1 j L 2 j L 3 j we wll also consder the 6 hyperplanes L(, j, k) := L 1 L 2 j L 3k where {, j, k} ={1, 2, 3}. Choose K 1 so that C L 21 =2. From (1), we have that ( C L 21, C L 22, C L 23 ) = (2, 9, 9). Moreover, as mentoned earler, we have { C L 31, C L 32, C L 33 } {{2, 9, 9}, {6, 6, 8}}. Suppose frst that ( C L 31, C L 32, C L 33 ) = (6, 6, 8). Then consderng L(1, 2, 3) we see that C L 11 3and by (1) equalty cannot hold, thus C L Smlarly, consderng K 3 shows that C L 13 2, and thus C L Thus the hyperplane K 2 ether contans at least 21 elements of C or meets C n 20 ponts ncludng exactly 5 ponts lyng n the hyperplane L 12 of K 2, contradctng (1). The cases ( C L 31, C L 32, C L 33 ) {(8,6,6), (6, 8, 6)} are smlar. On the other hand f ( C L 31, C L 32, C L 33 ) = (2, 9, 9), usng L(1, 2, 3), L 2 and L 3 we see that C L 1 j 2for j = 1, 2, 3 a contradcton wth C H 1 =9. The cases ( C L 31, C L 32, C L 33 ) {(9,2,9), (9, 9, 2)} are smlar. COROLLARY 3 The STS(243) AG(5, 3) cannot be 5-coloured. Proof. Assume that AG(5, 3) = C 1 C 2 C 3 C 4 C 5 s a 5-colourng of AG(5, 3) wth C 1 C for = 2, 3, 4, 5. Snce AG(5, 3) =243, we see that C 1 s a cap contanng at least 243 =49 ponts. 5 THEOREM 4 AG(5, 3) s a 6-chromatc STS(243). Proof. The mathematcal software MapleV was used to dscover the followng three parwse dsjont 45-caps C 1, C 2 and C 3 of AG(5, 3). Then a computer program was used to 3-colour the partal STS obtaned by restrctng AG(5, 3) to the 108 = 243 3(45) remanng ponts. We obtaned the followng 6-colourng of AG(5, 3). C 1 :={02201, 02101, 12202, 01211, 21111, 00120, 10221, 00112, 21100, 20210, 11211, 00002, 02020, 10020, 21000, 00010, 11110, 21210, 20120, 11121,00212, 00201, 22220, 02220, 20001, 22001, 21221, 21101, 10112, 22222, 22212, 00110, 02021, 22121, 10111, 21220, 01210, 02102, 20100, 01102, 01110, 22021, 02200, 11221, 22101}; C 2 :={00200, 11112, 10102, 01120, 00222, 20121, 00100, 11101, 02222, 11212, 22012, 20022, 22200, 12220, 22211, 02221, 01202, 10212, 22022, 21122, 22122, 21201, 22210, 02120, 10011, 01201, 00111, 20111, 02011, 21211, 02211, 01101, 00101, 00001, 20201, 21121, 21021, 10100, 00020, 22112, 02012, 21212, 21102, 11202, 02122}; C 3 :={02000, 20010, 22202, 10121, 10002, 12211, 22000, 11122, 21110, 02111, 02202, 21202, 20102, 02212, 11200, 02110, 22100, 01220, 00122, 11220, 01122, 22110, 01222, 21012, 22201, 21120, 00022, 21200, 00102, 11100, 10200, 22010, 00210, 02210, 10120, 00011, 01111, 00121, 00221, 22221, 02002, 20112, 21112, 20222, 21222}; C 4 :={11021, 21002, 01022, 10022, 21022, 12120, 02121, 12212, 11020, 20020, 22120,

5 CAPS AND COLOURING STEINER TRIPLE SYSTEMS , 10000, 12002, 01000, 12000, 12100, 01200, 11120, 01112, 11010, 20220, 01002, 10122, 20021, 12022, 02010, 11201, 10201, 10202, 12101, 20110, 00202, 22002, 20212, 12222}; C 5 :={20000, 11000, 02100, 12110, 10210, 11210, 10001, 01001, 21001, 12001, 10101, 12201, 01011, 21011, 12011, 22011, 00211, 10211, 01121, 20221, 12221, 20002, 11002, 11102, 12102, 22102, 20202, 00012, 10012, 01012, 12012, 12112, 01212, 11022, 02022, 20122}; C 6 :={01100, 20200, 12200, 10010, 01010, 21010, 12010, 10110, 12210, 01020, 21020, 12020, 22020, 00220, 10220, 11001, 02001, 20101, 20011, 11011, 11111, 12111, 22111, 20211, 00021, 10021, 01021, 12021, 12121, 01221, 20012, 11012, 02112, 12122, 10222, 11222}. It s shown n [3] that f v 3 (mod 6) and f there exsts an r-chromatc STS(v), then there exsts an r-chromatc STS(u) for every admssble u 2v + 1. Combnng ths fact wth Theorem 4 we get COROLLARY 5 There exsts a 6-chromatc STS(u) for every admssble u 487. Acknowledgments We thank Jean Fugère for hs help wth the computer program used to fnd a 6-colourng of AG(5, 3). References 1. M. de Brandes, K. T. Phelps and V. Rödl, Colorng Stener trple systems, SIAM J. Alg. Dsc. Math., Vol. 3 (1982) pp T. C. Brown and J. P. Buhler, A densty verson of a geometrc Ramsey theorem, J. Combnat. Theory A, Vol. 32 (1982) pp J. Fugère, L. Haddad and D. Wehlau, 5-chromatc STS s. JCD, Vol. 2, No. 5 (1994) pp D. G. Glynn, A lower bound for maxmal partal spreads n PG(3, q), Ars. Comb., Vol. 13 (1982) pp R. Hll, On Pellegrno s 20-caps n S 4,3. Combnatorcs, Vol. 81 (1981) pp , North-Holland Math. Stud., 78 North-Holland, Amsterdam-New York (1983). 6. R. Hll, Caps and codes, Dscrete Math., Vol. 22 (1978) pp L. Haddad, On the chromatc numbers of Stener trple systems. Preprnt, Dept. of Math & CS, RMC (1994). 8. H. Lenz and H. Zetler, Arcs and ovals n Stener trple systems, n Combnat. Theory. Proc. Conf. Schloss Rauschholzhausen (1982), Lec. Notes Math., Vol Sprnger, Berln, pp G. Pellegrno, Sul massmo ordne delle calotte n S 4,3, Matematche, Vol. 25 (1971) pp A. Street and D. Street, Combnatorcs of Expermental Desgn, Clarendon Press, Oxford (1987).