List coloring hypergraphs

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1 Lit coloring hypergraph Penny Haxell Jacque Vertraete Department of Combinatoric and Optimization Univerity of Waterloo Waterloo, Ontario, Canada Department of Mathematic Univerity of California San Diego, CA Submitted: 2010; Accepted: 2010; Publihed: XX Mathematic Subject Claification: 05C15, 05C65 Abtract Let H be a hypergraph and let L v : v V H) be et; we refer to thee et a lit and their element a color. A lit coloring of H i an aignment of a color from L v to each v V H) in uch a way that every edge of H contain a pair of vertice of different color. The hypergraph H i k-lit-colorable if it ha a lit coloring from any collection of lit of ize k. The lit chromatic number of H i the minimum k uch that H i k-lit-colorable. In thi paper we prove that every d-regular three-uniform linear hypergraph ha lit chromatic number at leat log d 5 log log d )1/2 provided d i large enough. On the other hand there exit d-regular three-uniform linear hypergraph with lit chromatic number at mot log 3 d + 3. Thi leave the quetion open a to the exitence of uch hypergraph with lit chromatic number olog d) a d. 1 Introduction A hypergraph H i k-uniform if every edge of H ha ize k, and d-regular if every vertex of H i in exactly d edge of H. A hypergraph i linear if any pair of ditinct edge of the hypergraph interect in at mot one vertex. Let H be a hypergraph and Partially upported by NSERC Reearch upported by an Alfred P. Sloan Reearch Fellowhip and NSF Grant DMS the electronic journal of combinatoric ), #R00 1

2 let L v : v V H) be et; we refer to thee et a lit. A lit coloring of H i an aignment of an element of L v to each v V H) in uch a way that every edge of H contain a pair of vertice aigned different element. The hypergraph H i called k-lit-colorable if it ha a lit coloring from any collection of lit of ize k. The lit chromatic number χ l H) of H, alo called the choice number of H, i the minimum k uch that H i k-lit-colorable. In thi paper, we tudy the lit chromatic number of linear uniform regular hypergraph. 1.1 Lit coloring graph The notion of lit-coloring i a generalization of the notion of proper coloring, and ha been tudied extenively for graph. In particular, Alon [1] howed that every bipartite graph of minimum degree at leat d ha lit chromatic number at leat 1 2 log 2 d, improving a preceding lower bound of order log d log log d a d in [2]. On the other hand it i known [9] that the complete bipartite graph K d,d with d vertice in each part atifie χk d,d ) = 1+o1)) log 2 d. It ha been aked whether every bipartite graph with maximum degree d ha lit chromatic number Olog d) ee Alon and Krivelevich [4]) but thi tantalizing problem remain open. Here and in what follow, the logarithm i taken to be the natural logarithm unle a bae i explicitly diplayed. 1.2 Lit coloring hypergraph There eem to be very few reult on lit coloring of hypergraph. Perhap the mot famou quetion on coloring of hypergraph i the Erdő-Faber-Lováz conjecture ee [6]): given an n-uniform linear hypergraph coniting of n edge, there i vertex-coloring of the hypergraph for which every edge receive all n color. Equivalently, the conjecture tate that any graph compriing the union of n edge-dijoint clique of ize n ha chromatic number n. Via a more general reult on coloring linear hypergraph, Kahn [11] howed that that an n-uniform linear hypergraph can be vertex-colored with n + on) color in uch a way that the vertice in each edge all receive different color. Recently, a imple proof of the Erdő-Faber-Lováz conjecture wa announced. For Steiner triple ytem with n vertice three-uniform hypergraph in which every pair of vertice i covered exactly once a lower bound of order log n/ log log n for the lit chromatic number wa hown in [10]. In thi paper, we concentrate on giving bound on the lit chromatic number of all linear regular three-uniform hypergraph, which we refer to a triple ytem. We hall check in thi paper that the complete r-partite r-uniform hypergraph K r n with part of ize n ha lit chromatic number aymptotic to log r n a the electronic journal of combinatoric ), #R00 2

3 n. 1.3 Main Theorem The main reult of thi paper how that every d-regular linear triple ytem ha large lit chromatic number: Theorem 1 There exit a contant d 0 uch that for d d 0, every d-regular linear triple ytem H ha χ l H) > log d ) 1/2. 5 log log d On the other hand, there exit a d-regular linear triple ytem with lit chromatic number at mot log 3 d + 3. The requirement of linearity in thi theorem i neceary. Conider the triple ytem coniting of vertex et V = V 1 V n W with V i = 2 and W = n and where the edge et E conit of {e V : e W = 1, i : V i e}. Thi i an n-regular triple ytem whoe lit chromatic number i two, ince for any aignment of lit we can chooe different color for the two vertice inide each V i. Relative to the theorem above, the mot relevant open quetion i the exitence of d- regular linear triple ytem with lit chromatic number olog d) a d. We leave thi a an open problem. Problem. Do there exit three-uniform d-regular linear hypergraph of lit chromatic number olog d) a d? 2 Lit coloring complete r-partite hypergraph In thi ection, we how that for infinitely many d, there exit a linear d-regular triple ytem with lit chromatic number at mot log 3 d + 3. The example i the following three-partite triple ytem K 3 d L): if L i any d d latin quare, then we create a linear triple ytem with three part a follow. We index the row of L by a et R of ize d, the column by a et C of ize d, and then let R, C, [d] denote the part of K 3 d L). Then the edge of K 3 d L) are all triple {r, c, i} with r R, c C, and uch that i i in poition r, c) in L. Note that by definition of a latin quare there are no repeated entrie in any row or column of L, o the triple ytem K 3 d L) i d-regular and linear. We will how that it ha lit chromatic number at mot log 3 d+3 by howing that the electronic journal of combinatoric ), #R00 3

4 the complete r-partite hypergraph K r d which contain K 3 d L) when r = 3 ha lit chromatic number at mot log r d + 3. We will alo how that K r d ha lit chromatic number at leat log r d Olog log d) a d. Our argument i baed on that of [9]. We define a hypergraph H to be r-colorable if the lit aignment in which all lit are {1,..., r} admit a lit coloring of H. The following lemma will be ued: Lemma 2 Let m r k) denote the minimum number of edge in a k-uniform hypergraph which i not r-colorable. Then a k, r k 1 m r k) O r k 2 r k ). Proof A random r-coloring of a hypergraph conit in chooing uniformly at random a color from [r] independently for each vertex in the hypergraph. The lower bound of r k 1 in the lemma follow from the fact that in a random r-coloring of a k-uniform hypergraph with m edge, the expected number of edge whoe vertice are all aigned the ame color i r 1 k m, o if m < r k 1, then the hypergraph i r-colorable. In fact, thi i all we hall need to prove χ l K r n ) log r n + 3. The upper bound on m r k) alo follow from probabilitic method; we ketch the proof for r > 2, ince it i very imilar to the proof for r = 2 given in [7] ee alo Alon and Spencer [5], Page 9). We conider a hypergraph H with vertex et [rk 2 ] contructed by randomly and uniformly electing et of ize k in T independent round from [rk 2 ]. We hall take T = 2k 2 er) k log r. An r-coloring of H i a partition X 1, X 2,..., X r ) of the vertex et [rk 2 ] of H. An edge of H i monochromatic if it lie entirely in ome X i. We ay that an r-coloring of H fail in round i if in the ith round, the choen et of ize k i monochromatic. The aim i to how that the expected number of r-coloring which do not fail in any round i le than one, and we follow the computation of Erdő [7]. The probability that the edge choen at round i i monochromatic for a given coloring X 1, X 2,..., X r ) i exactly ) rk 2 1 r ) Xi. k k i=1 By convexity of binomial coefficient, the um i a minimum when X 1 = X 2 = = X r = k 2 : r ) ) Xi k 2 r. k k i=1 the electronic journal of combinatoric ), #R00 4

5 Uing the tandard bound ) rk 2 k erk) k and ) k 2 k k k, the probability that an edge choen at round i i monochromatic i at leat r/er) k. It follow that the probability that a given coloring X 1, X 2,..., X r ) doe not fail at any round i at mot 1 r er) k ) T ince the round are independent. Now T wa choen o that thi quantity i le than r rk2. Since the number of r-coloring of [rk 2 ] i r rk2, we conclude that the expected number of r-coloring which do not fail at any round i le than one. In particular, there i a witne H to thi event. Thi hypergraph H ha exactly T edge and i not r-colorable, a required. We ue the bound on m r k) in the above lemma to give bound on χ l K r m ) a follow: Theorem 3 For all r 2 and a m, χ l K r m ) = 1 + o1)) log r m. Proof If we can how χ l K r m ) k when rm = m r k) 1 and χ l K r m ) k+1 when m = m r k), then we are done uing the bound in the lat lemma, ince rm + 1 r k 1 implie k log r m + 3 and m Ck 2 r k implie k log r m 2 log r log r m O1) when m, a required. Firt we how that for m = m r k), we have χ l K r m ) > k. By definition of m r k) = m, there i a k-uniform hypergraph F with m edge uch that F i not r-colorable. We let the lit in the ith part V i of K r m be exactly the edge of F, for 1 i r. We claim there i no coloring of K r m from thi lit aignment. Suppoe, for a contradiction, that there i a lit coloring of K r m from thee lit, and let S 1, S 2,..., S r be the et of color ued on the vertice of V 1, V 2,..., V r, repectively. Note that each S i i actually a et of vertice of F. We oberve that r S i = i=1 otherwie a certain color appear on a vertex v i V i for 1 i r, in which cae {v 1, v 2,..., v r } i a monochromatic edge of K r m. Since each S i i a tranveral of the edge in F, T i = V F )\S i doe not contain any edge of F for 1 i r. Since S i =, every vertex of F i in the complement of ome S i and therefore T 1 T r = V F ). Thi contradict that F i not r-colorable, and hence χ l K r,m ) > k when m = m r k). Converely we how χ l K r m ) k when rm = m r k) 1. Conider an aignment of lit of ize k to the rm vertice of K r m, and let F be the k-uniform hypergraph of all the electronic journal of combinatoric ), #R00 5

6 thoe lit. Then F ha an r-coloring ince F < m r k). Fix an r-coloring of F, ay V F ) = T 1 T 2 T r. The lit L v at a vertex v V i mut contain an element not in T i, otherwie L v would be monochromatic under the coloring T 1 T 2 T r of F. To each v V i we aign an arbitrary element of L v \T i, for 1 i r. We claim thi i a proper coloring of K r m. If not, then there i an edge {v 1, v 2,..., v r } in K r m where v i V i, and uch that every v i : 1 i r receive the ame color, which i an element of T j for ome j [r]. However, v j wa aigned an element of L vj \T j, which i a contradiction. Therefore χ l K r m ) k. Thi complete the proof. It appear to be an intereting quetion to determine fd) = min L χ l K 3 d L)) where the minimum i over all d d latin quare. In particular, it would be intereting to determine whether fd) = olog d) a d. 3 Lemma The probabilitic lemma we ue to prove Theorem 1 are given here. The firt lemma i the Chernoff bound, one of the baic tool in probabilitic method ee for example Alon and Spencer [5]. Lemma 4 Let Z 1, Z 2,..., Z n be identically ditributed independent random variable where P Z i = 1) = p and P Z i = 0) = 1 p, and let S be their um. Then for any ϵ 0, 1], P S ES) > ϵes)) 2 exp ϵ 2 ES)/2). We alo require the Lováz Local Lemma [8] in the following form. Here and in what follow, A c denote the complement of an event A. Lemma 5 Let A 1, A 2,..., A n be event in a probability pace, uch that each A i i mutually independent of any ubet of event indexed by the et [n]\j i for ome dependency et J i [n] where max J i =. Suppoe that P A i ) 1 4 for all i [n]. Then n P i=1 A c i) exp n ). the electronic journal of combinatoric ), #R00 6

7 4 Proof of Theorem 1 We are given a d-regular linear triple ytem H, and we want to come up with a collection of lit on the vertice from which no coloring i poible and where the lit are a large a poible. The idea of the proof i to how that if we aign random lit of length in [t] := {1, 2,..., t} to the vertice of H, for ome carefully choen value of and t, then with poitive probability there i no proper coloring of H from thee lit. We will chooe = log d) 1/2 5 log log d) 1/2 and t = 8) +4) and put p := 1/8) 3 t. If d i a large enough contant, then it i traightforward to check that the parameter p,, t atify for completene a verification i written in the appendix): t exp 2 p 3 d/4t 2 ) < 1/64d 2 1) exp n/8) 2 t) < 3pn 2) exp p 2 n/8) < exp n/d 2 ). 3) For any K 8), we record that tandard bound on binomial coefficient ee Appendix : Lemma 6) give ) ) t/k t > 2K). 4) 4.1 Preproceing Define random et X V H) and Y V H) by independently placing each vertex of V H) into X with probability p 2 and into Y with probability p and into Z = V H)\X Y ) with probability 1 p p 2. Let H denote the three-partite hypergraph coniting of all {x, y, z} H uch that x X, y Y and z Z. We write dx, y) > 2 to denote that in H every edge on x i dijoint from every edge on y in other word x and y are at ditance more than two. For y Y let Γ X y) = {x X : {x, y, z} H for ome z} and for z Z let Γ XY z) = {x, y) X Y : {x, y, z} H }. Define d X y) = Γ X y) and d XY z) = Γ XY z). Since H i a d-regular linear hypergraph, we have for y Y and z Z, Ed X y)) = 2p 2 1 p p 2 )d and Ed XY z)) = 2p 3 d. Let A y : y Y be the event p 2 d < d X y) < 4p 2 d and let A z : z Z be the event p 3 d < d XY z) < 4p 3 d. the electronic journal of combinatoric ), #R00 7

8 Claim 1. With poitive probability, each event A X = { 1 2 p2 n < X < 2p 2 n} and A Y = { 1 2 pn < Y < 2pn} and every A y : y Y and every A z : z Z occur. Proof. We hall apply the Chernoff Bound with ϵ = 1/ 8. The event A y contain the event d X y) Ed X y)) < ϵed X y)), ince p < 1/8 3, and clearly A z contain the event d XY z) Ed XY z)) < ϵed XY z)). Since vertice are placed independently in the et X and Y, Lemma 4 how P A c y) 2 exp p2 d ) 8 and P A c z) 2 exp p3 d ) 8 and imilarly P A c X ) 2 exp p2 n/8) and P A c y) 2 exp pn/8). A dependency graph of the event A y : y Y and A z : z Z ha maximum degree at mot = 4d 2, ince any ingle event A c v i mutually independent of any et of event A c w : dv, w) > 2. By 1), both the bound in 5) are eaily le than 1/4. By Lemma 5, with probability at leat exp n/4d 2 ) every A y : y Y and every A z : z Z occur. By 3), exp n/4d 2 ) > P A c X Ac Y ), and o with poitive probability, the event A X and A Y and every A y : y Y and A z : z Z occur. Thi prove the claim. For the remainder of the proof, we work in a ubhypergraph H for which all the event in Claim 1 hold, and we aume that X, Y and Z are the part of H. 5) 4.2 Choice of lit in X Firt we aign lit to X. We chooe uniformly and independently random lit of ize from [t] for the vertice of X. For y Y, let B y be the event that no color appear in more than 2d X y)/t lit in Γ X y) and for z Z let B z be the event that no color appear in more than 2d XY z)/t lit in Γ X z) := {x X : y Y, x, y) Γ XY z)}. Claim 2. With poitive probability, every B y : y Y and every B z : z Z occur. Proof. Since H i linear, note that Γ X z) = d XY z). The expected number of time a particular color appear in lit in Γ X y) i exactly d X y)/t and the expected number of time a particular color appear in Γ X z) i d XY z)/t. Since the lit are choen independently, the probability that a particular color appear in more than 2d X y)/t lit in Γ X y) i at mot 2 exp d X y)/2t), by the Chernoff Bound with ϵ = 1. A imilar tatement hold for the color in Γ X z), and ince there are at mot t color, we the electronic journal of combinatoric ), #R00 8

9 deduce from the union bound that P B c y) < 2t exp d X y)/2t) < 2t exp p 2 d/2t) P B c z) < 2t exp d XY z)/2t) < 2t exp p 3 d/2t). A dependency graph of the event B c v ha maximum degree at mot = 4d 2, ince Bv c i mutually independent of any event Bw c : dv, w) > 2. By 1), we eaily have P By) c < 1/4 and P Bz) c < 1/4, and o Lemma 5 complete the proof of Claim 2. From now on we fix an aignment of lit L of ize from [t] to the vertice of X, uch that every B y : y Y and every B z : z Z occur. Let ρ y be the number of color ued at leat d X y)/2t time on Γ X y). Since B y occur, t ρ y ) d Xy) 2t + ρ y 2d X y) t d X y) and it follow that ρ y > t/4. Since every B y occur, we have that for each y Y and each coloring of X from L there exit a et S y of t/4 color each appearing at leat p 2 d/2t time in Γ X y). 4.3 Choice of lit in Y Now we independently and randomly aign lit L y of ize from [t] to the y Y. For a fixed coloring of X from L, if L y S y, then any color elected from L y reult in at leat p 2 d/2t vertice x Γ X y) of the ame color a y. Let B χ be the event that under a coloring χ of X, L y S y for at leat 1 2 8) Y vertice y Y, and let B = χ B χ where the interection i over all coloring of X from L. For z Z, let C z be the event that no color appear on the lit at x and at y for at leat 4 2 d XY z)/t 2 pair x, y) Γ XY z). Claim 3. With poitive probability, B a well a every C z : z Z occur. Proof. We oberve that for every χ and every y, ) ) t/4 t P L y S y ) / > 8) uing 4) with K = 4. Therefore the expected number of event L y S y i at leat 8) Y. Since the L y are choen independently, the Chernoff Bound with ϵ = 1/2 how that for a fixed coloring χ of X, P Bχ) c < 2 exp Y ). 88) the electronic journal of combinatoric ), #R00 9

10 Since there are at mot X coloring of X, and ince by Claim 1 Y pn/2 and X 2p 2 n, the expected number of coloring χ of X for which B c χ occur i at mot 2 exp Y ) X < 2 exp pn ) 88) 168) + 2p2 n log < exp pn ). 208) By 2), with room to pare, thi i le than exp n/d 2 ), and o Markov Inequality give P B c ) < exp n/2d 2 ). For x Γ X z), there i a unique y Γ Y z) uch that x, y) Γ XY z), ince H i linear. The chance that y i aigned a lit containing a particular color in L x i exactly /t. Fix a color i, and let C i z be the event that color i appear on the lit at x and at y for at leat 4 2 d XY z)/t 2 pair x, y) Γ XY z). Recall that ince every B z occur, color i appear at mot 2d XY z)/t time in Γ X z). The expected number of x, y) Γ XY z) uch that i L x L y i at mot 2 2 d XY z)/t 2. Since lit are aigned independently, we may apply the Chernoff Bound with ϵ = 1 to obtain P C i z) < 2 exp 2 d XY z)/t 2 ). Since there are t color in total, the union bound how By Claim 1, thi i at mot P C c z) < 2t exp 2 d XY z)/t 2 ). t exp 2 p 3 d/2t 2 ). Now C c z i mutually independent of any event C c w : dw, z) > 2, o a dependency graph of event C c z ha maximum degree at mot = 4d 2. By 1), P C c z) < 1/4 and o Lemma 5 how that with probability at leat exp n/4d 2 ), every C z occur. Uing the preceding bound on P B c ), we ee that with poitive probability every C z : z Z and B occur. Let u now extend our lit aignment L to include a lit aignment of element from [t] to each y Y, uch that B and each C z : z Z occur. 4.4 Choice of lit for Z We have aigned lit L to the vertice of X and Y in uch a way that in every coloring of X, there i a et S Y of ize at leat pn/48) with the property that for y S, L y S y thi i the event B), and for every z Z, every color appear on the lit at x and at y for at mot 4 2 d XY z)/t 2 pair x, y) Γ XY z) thee are the event C z ). Now we how that there exit an aignment of lit to Z from which no coloring of H i poible. the electronic journal of combinatoric ), #R00 10

11 Let G denote the bipartite graph coniting of part X and Y and edge z Z Γ XY z). If κ i any fixed coloring of X Y, then ince B occur, whichever color from L y that i aigned to y by κ reult in at leat p 2 d/2t pair x, y) G uch that x and y have the ame color. Therefore we have at leat y S p 2 d 2t > p3 dn 8t8) pair x, y) G uch that x and y are aigned the ame color by κ. We call thee monochromatic edge of G. Let T denote the et of z Z uch that Γ XY z) contain at leat p 3 d/t8) +1 monochromatic edge under κ. Since every C z occur, no Γ XY z) can contain more than t 4 2 d XY z)/t p 3 d/t monochromatic edge under κ. Therefore T 162 p 3 d t + n T ) p 3 d t8) +1 p3 dn 4t8). From thi it follow that T > n/8) +2. Since every C z occur, for every z T, Γ XY z) contain monochromatic edge of at leat p 3 d t8) +1 t p 3 d > 2t 8) +3 different color under κ. Let T z,κ be thi et of color on monochromatic edge of Γ XY z). We aign random lit to Z of ize from [t]. In order for H, and therefore H, to have a lit coloring extending κ, it cannot be that ome lit at a vertex z T i contained in the et T z,κ. Let C κ be the event that no lit at any vertex z T i contained in T z,κ. By 4) with K = 8) +3 /2, and therefore P C κ ) < 1 ) 2t/8) +3 2t/8) +3 t ) > 1 2K) ) t )) T < exp T ) < exp n ) 2K) 8) 2 t uing that T > n/8) +2. By 2), P C κ ) < 3pn, and ince there are at mot X Y 3pn coloring κ of X Y, we deduce that there exit a lit aignment to the vertice of Z for which no C κ occur. In word, there i an aignment of lit to the vertice of Z uch that for any coloring κ of X Y, ome z Z cannot be properly colored from it lit. For thi lit aignment, no coloring of H exit. Thi complete the proof. the electronic journal of combinatoric ), #R00 11

12 5 Note added in proof We have learned that the lit coloring problem for linear and uniform hypergraph ha been tudied independently by Alon and Kotochka [3]. They prove a more general verion of Theorem 1, in that they conider r-uniform hypergraph for r 3, and they do not retrict to d-regular hypergraph but give bound in term of the average degree d. When pecialied to the etting of Theorem 1, their reult give imilar bound, and in particular we till do not know whether there exit 3-uniform d-regular hypergraph with lit chromatic number olog d) a d. 6 Appendix We choe = log d) 1/2 5 log log d) 1/2 and t = 8) +4) and put p := 1/8) 3 t. If d i a large enough contant, we claim that the following three inequalitie hold: t exp 2 p 3 d/4t 2 ) < 1/64d 2 6) exp n/8) 2 t) < 3pn 7) exp p 2 n/8) < exp n/d 2 ). 8) To verify the inequalitie 6), 7), 8) we firt oberve that with the given definition we have 22 < d 1/5 and o 2 < d 1/10. Thu when d i large enough we find t = 8) +4) < d 1/9. Therefore to prove 6) we note that for large enough d, t exp 2 p 3 d/4t 2 ) < d 1/9 exp 2 d/48) 9 t 5 ) < d 1/9 exp d 1/3 ) < 1/64d 2. For 7) it uffice to how 3p log < 1/64 2 t, which i immediate from the definition of p. Finally for 8) we want 1/d 2 < p 2 /8, o ince p 2 = 1/8) 6 t 2 > 1/d 1/3 the inequality hold provided d 5/3 > 8. Lemma 6 Let, t be poitive integer and uppoe that t/k 2. Then ) ) t/k t > 2K). Proof By definition we have ) t/k > On the other hand and the reult follow. t/k )! ) t t! > 1! t ). 2K the electronic journal of combinatoric ), #R00 12

13 Acknowledgment Thi work wa done while the author were viiting the Intitute for Pure and Applied Mathematic at the Univerity of California, Lo Angele. The author wih to thank IPAM for their upport. Reference [1] Alon, N., Degree and choice number, Random Structure & Algorithm ), [2] Alon, N., Retricted coloring of graph. In: K. Walker, Editor, Survey in Combinatoric, Proc 14th Britih Combinatorial Conference, London Mathematical Society Lecture Note Serie 187, Cambridge Univ. Pre 1993), [3] Alon, N., Kotochka, A, Hypergraph lit coloring and Euclidean Ramey theory, ubmitted, [4] Alon, N., Krivelevich, M., The choice number of random bipartite graph, Annal of Combinatoric ), [5] Alon, N., Spencer, J. The Probabilitic Method, Wiley-Intercience, [6] Erdő, P., On the combinatorial problem I would mot like to ee olved, Combinatorica ), [7] Erdő, P., On a combinatorial problem. II. Acta Mathematica Academiae Scientiarum Hungaricae ), [8] Erdő, P., Lováz, L., Problem and reult on 3-chromatic hypergraph and ome related quetion, in Infinite and Finite Set A. Hajnal et al, ed.), North-Holland, Amterdam 1975), [9] Erdő, P., Rubin, A., Taylor, H., Chooability in graph, Proc. Wet Coat Conf. on Combinatoric, Graph Theory and Computing, Congreu Numerantium ), [10] Haxell, P., Pei, M., On lit coloring Steiner triple ytem, Journal of Combinatorial Deign ), [11] Kahn, J., Coloring nearly-dijoint hypergraph with n + on) color, Journal of Combinatorial Theory A ), the electronic journal of combinatoric ), #R00 13