Nonnegative k-sums, fractional covers, and probability of small deviations

Transcription

1 Noegative k-sums, fractioal covers, ad probability of small deviatios Noga Alo Hao Huag Bey Sudakov Abstract More tha twety years ago, Maickam, Miklós, ad Sighi cojectured that for ay itegers, k satisfyig 4k, every set of real umbers with oegative sum has at least 1 k- elemet subsets whose sum is also oegative I this paper we discuss the coectio of this problem with matchigs ad fractioal covers of hypergraphs, ad with the questio of estimatig the probability that the sum of oegative idepedet radom variables eceeds its epectatio by a give amout Usig these coectios together with some probabilistic techiques, we verify the cojecture for 33k 2 This substatially improves the best previously kow epoetial lower boud e ck log log k I additio we prove a tight stability result showig that for every k ad all sufficietly large, every set of reals with a oegative sum that does ot cotai a member whose sum with ay other members is oegative, cotais at least subsets of cardiality k with oegative sum 1 Itroductio Let { 1,, } be a set of real umbers whose sum is oegative It is atural to ask the followig questio: how may subsets of oegative sum must it always have? The aswer is quite straightforward, oe ca set 1 = 1 ad all the other i = 1, which gives 2 1 subsets This costructio is also the smallest possible sice for every subset A, either A or []\A or both must have a oegative sum Aother atural questio is, what happes if we further restrict all the subsets to have a fied size k? The same eample yields 1 oegative k-sums cosistig of 1 ad 1 s This costructio is similar to the etremal eample i the Erdős-Ko-Rado theorem [8] which states that for 2k, a family of subsets of size k i [] with the property that every two subsets have a oempty itersectio has size at most 1 However the relatio betwee k-sum ad k-itersectig family is somewhat subtle ad there is o obvious way to traslate oe problem to the other Deote by A, k the miimum possible umber of oegative k-sums over all possible choices of umbers 1,, with i=1 i 0 For which values of ad k, is the costructio 1 = Sackler School of Mathematics ad Blavatik School of Computer Sciece, Tel Aviv Uiversity, Tel Aviv 69978, Israel ogaa@tauacil Research supported i part by a ERC Advaced grat ad by a USA-Israeli BSF grat Departmet of Mathematics, UCLA, Los Ageles, CA, huaghao@mathuclaedu Departmet of Mathematics, UCLA, Los Ageles, CA bsudakov@mathuclaedu Research supported i part by NSF grat DMS , NSF CAREER award DMS ad by USA-Israeli BSF grat 1

2 1, 2 = = = 1 best possible? I other words, whe ca we guaratee that A, k = 1? This questio was first raised by Bier ad Maickam [4, 5] i their study of the so-called first distributio ivariat of the Johso scheme I 1987, Maickam ad Miklós [15] proposed the followig cojecture, which i the laguage of the Johso scheme was also posed by Maickam ad Sighi [16] i 1988 Cojecture 11 For all 4k, we have A, k = 1 I the Erdős-Ko-Rado theorem, if < 2k, all the k-subsets form a itersectig family of size k > 1 But for > 2k, the star structure, which always takes oe fied elemet ad other arbitrarily chose elemets, will do better tha the set of all k-subsets of the first 2 elemets For a similar reaso we have the etra coditio 4k i the Maickam-Mikós-Sighi cojecture 1 is ot the best costructio whe is very small compared to k For eample, take = 3k + 1 umbers, 3 of which are equal to 3k 2 ad the other 3k 2 umbers are 3 It is easy to see that the sum is zero O the other had, the oegative k-sums are those subsets cosistig oly of 3 s, which gives 3k 2 oegative k-sums It is ot difficult to verify that whe k > 2, 3k 2 k k < 3k+1 1 However this kid of costructio does ot eist for larger The Maickam-Mikós-Sighi cojecture has bee ope for more tha two decades Oly a few partial results of this cojecture are kow so far The most importat oe amog them is that the cojecture holds for all divisible by k This claim ca be proved directly by cosiderig a radom partitio of our set of umbers ito pairwise disjoit sets, each of size k, but it also follows immediately from Barayai s partitio theorem [2] This theorem asserts that if k, the the family of all k-subsets of [] ca be partitioed ito disjoit subfamilies so that each subfamily is a perfect k-matchig Sice the total sum is oegative, amog the /k subsets from each subfamily, there must be at least oe havig a oegative sum Hece there are o less tha k //k = 1 oegative k-sums i total Besides this case, the cojecture is also kow to be true for small k It is ot hard to check it for k = 2, ad the case k = 3 was settled by Maickam [14], ad by Mario ad Chiaselotti [17] idepedetly Let fk be the miimal umber N such that A, k = 1 for all N The Maickam- Miklós-Sighi cojecture states that fk 4k The eistece of such fuctio f was first demostrated by Maickam ad Miklós [15] by showig fk k k + k 2 + k Bhattacharya [3] foud a ew ad shorter proof of eistece of f later, but he did t improve the previous boud Very recetly, Tyomky [20] obtaied a better upper boud fk k4e log k k e ck log log k, which is still epoetial I this paper we discuss a coectio betwee the Maickam-Miklós-Sighi cojecture ad a problem about matchigs i dese uiform hypergraph We call a hypergraph H r-uiform if all the edges have size r Deote by νh the matchig umber of H, which is the maimum umber of pairwise disjoit edges i H For our applicatio, we eed the fact that if a -uiform hypergraph o 1 vertices has matchig umber at most /k, the its umber of edges caot eceed c 1 for some costat c < 1 idepedet of, k This is closely related to a special case of a log-stadig ope problem of Erdős [7], who i 1965 asked to determie the maimum possible umber of edges of a r-uiform hypergraph H o vertices with matchig umber νh Erdős cojectured that the 2

3 optimal case is whe H is a clique or the complemet of a clique, more precisely, for νh < /r the maimum possible umber of edges is give by the followig equatio: { r[νh } + 1] 1 νh ma eh = ma, 1 r r r For our applicatio to the Maickam-Miklós-Sighi cojecture, it suffices to prove a weaker statemet which bouds the umber of edges as a fuctio of the fractioal matchig umber ν H istead of νh To attack the latter problem we combie duality with a probabilistic techique together with a iequality by Feige [9] which bouds the probability that the sum of a arbitrary umber of oegative idepedet radom variables eceeds its epectatio by a give amout Usig this machiery, we obtai the first polyomial upper boud fk 33k 2, which substatially improves all the previous epoetial estimates Theorem 12 Give itegers ad k satisfyig 33k 2, for ay real umbers { 1,, } whose sum is oegative, there are at least 1 oegative k-sums Recall that earlier we metioed the similarity betwee the Maickam-Miklós-Sighi cojecture ad the Erdős-Ko-Rado theorem Whe 4k, the cojectured etremal eample is 1 = 1, 2 = = = 1, where all the 1 oegative k-sums use 1 For the Erdős-Ko-Rado theorem whe > 2k, the etremal family also cosists of all the 1 subsets cotaiig oe fied elemet It is a atural questio to ask if this kid of structure is forbidde, ca we obtai a sigificat improvemet o the 1 boud? A classical result of Hilto ad Miler [12] asserts that if > 2k ad o elemet is cotaied i every k-subset, the the itersectig family has size at most 1 + 1, with the etremal eample beig oe of the followig two Fi [] ad X []\{}, X = k The family F 1 cosists of X ad all the k-subsets cotaiig ad itersectig with X Take Y [], Y = 3 The family F 2 cosists of all the k-subsets of [] which itersects Y with at least two elemets It ca be easily checked that both families are itersectig ad F 1 = 1 + 1, F 2 = 3 3 k k 3 Whe k = 3, F1 = F 2 ad their structures are o-isomorphic For k 4, F 1 > F 2, so oly the first costructio is optimal Here we prove a Hilto-Miler type result about the miimum umber of oegative k-sums Call a umber i large if its sum with ay other umbers j is oegative We prove that if o i is large, the the 1 boud ca be greatly improved We also show that there are two etremal structures, oe of which is maimum for every k ad the other oly for k = 3 This result ca be cosidered as a aalogue of the two etremal cases metioed above i the Hilto-Miler theorem Theorem 13 For ay fied iteger k ad sufficietly large, ad for ay real umbers 1,, with i=1 i 0, where o i is large, the umber N of differet oegative k-sums is at least

4 For large, Theorem 13 whose statemet is tight improves the 1 boud i the oegative k-sum problem to whe large umbers are forbidde This boud is asymptotically 2 + o1 1 Call a umber i 1 δ-moderately large, if there are at least 1 δ 1 oegative k-sums usig i, for some costat 0 δ < 1 I particular, whe δ = 0 this is the defiitio of a large umber If there is o 1 δ-moderately large umber for some positive δ, we ca prove a much stroger result assertig that at least a costat proportio of the k k-sums are oegative More precisely, we prove the followig statemet Theorem 14 There eists a positive fuctio gδ, k, such that for ay fied k ad δ ad all sufficietly large, the followig holds For ay set of real umbers 1,, with oegative sum i which o member is 1 δ-moderately large, the umber N of oegative k-sums i the set is at least gδ, k k The rest of this paper is orgaized as follows I the et sectio we preset a quick proof of a slightly worse boud for the fuctio fk defied above, amely, we show that fk 2k 3 The proof uses a simple estimate o the umber of edges i a hypergraph with a give matchig umber The proof of Theorem 12 appears i Sectio 3, where we improve this estimate usig more sophisticated probabilistic tools I Sectio 4 we prove the Hilto-Miler type results Theorem 13 ad 14 The last sectio cotais some cocludig remarks ad ope problems 2 Noegative k-sums ad hypergraph matchigs I this sectio we discuss the coectio of the Maickam-Miklós-Sighi cojecture ad hypergraph matchigs, ad verify the cojecture for 2k 3 Without loss of geerality, we ca assume i=1 i = 0 ad 1 2 with 1 > 0 If the sum of 1 ad the smallest umbers k+2,, is oegative, there are already 1 oegative k-sums by takig 1 ad ay other umbers Cosequetly we ca further assume that 1 + k < 0 As all the umbers sum up to zero, we have k+1 > 0 2 Let m be the largest iteger ot eceedig k which is divisible by k, the 2k + 1 m k Sice the umbers are sorted i descedig order, we have m+1 m k k+1 > 0 3 As metioed i the itroductio, the Maickam-Mikós-Sighi cojecture holds whe is divisible by k by Barayai s partitio theorem, thus there are at least m 1 2k oegative k-sums usig oly umbers from { 2,, m+1 } From ow o we are focusig o coutig the umber of oegative k-sums ivolvig 1 If this umber plus 2k is at least 1, the the Maickam- Miklós-Sighi cojecture is true Recall that i the proof of the case k, if we regard all the egative k-sums as edges i a k-uiform hypergraph, the the assumptio that all umbers add up to zero provides us the fact that 4

5 this hypergraph does ot have a perfect k-matchig Oe ca prove there are at least 1 edges i the complemet of such a hypergraph, which eactly tells the miimum umber of oegative k- sums We utilize the same idea to estimate the umber of oegative k-sums ivolvig 1 Costruct a -uiform hypergraph H o the verte set {2,, } The edge set EH cosists of all the -tuples {i 1,, i } correspodig to the egative k-sum 1 + i1 + + i < 0 Our goal is to show that eh = EH caot be too large, ad therefore there must be lots of oegative k-sums ivolvig 1 Deote by νh the matchig umber of our hypergraph H, which is the maimum umber of disjoit edges i H By defiitio, every edge correspods to a -sum which is less tha 1, thus the sum of the νh umbers correspodig to the vertices i the maimal matchig is less tha νh 1 O the other had, all the remaiig 1 νh umbers are at most 1 Therefore 1 = νh νh 1 By solvig this iequality, we have the followig lemma Lemma 21 The matchig umber νh is at most /k If the matchig umber of a hypergraph is kow ad is large with respect to k, we are able to boud the umber of its edges usig the followig lemma We deote by H the complemet of the hypergraph H Lemma 22 If > k 3, ay -uiform hypergraph H o 1 vertices with matchig umber 1 at most /k has at least 1 k+1 edges missig from it Proof Cosider a radom permutatio σ o the 1 vertices v 1,, v 1 of H Let the radom variables Z 1 = 1 if σv 1,, σv is a edge i H ad 0 otherwise Repeat the same process for the et idices ad so o Fially we will have at least m k radom variables Z 1,, Z m Let Z = Z Z m Z is always at most /k sice there is o matchig of size larger tha /k O the other had, EZ i is the probability that radomly chose vertices form a edge i H, therefore EZ i = eh/ 1 Hece, The umber of edges missig is equal to eh = 1 therefore k eh eh EZ = m 1 k eh 1 4 eh By 4, eh 1 1 1, = [ k k k 3 [ 1 1 k k k 3 k 1 ] 1 ] 1 k k Now we ca easily prove a polyomial upper boud for the fuctio fk cosidered i the itroductio, showig that fk 2k 3 5 5

6 Theorem 23 If 2k 3, the for ay real umbers { 1,, } whose sum is oegative, the umber of oegative k-sums is at least 1 1 Proof By Lemma 22, there are at least 1 k+1 edges missig i H, which also gives a lower boud for the umber of oegative k-sums ivolvig 1 Together with the previous 2k oegative 1 k-sums without usig 1, there are at least k+1 claim that this umber is greater tha 1 2k 1 + 2k oegative k-sums i total We whe 2k 3 This statemet is equivalet to provig / 1 1 1/k + 1, which ca be completed as follows: 2k 1 / = k k k 3 k k + 1 The last iequality is because 2k + 1 = 2k 3 k 2 2k + 1 2k 3 k Fractioal covers ad small deviatios The method above verifies the Maickam-Miklós-Sighi cojecture for 2k 3 ad improves the curret best epoetial lower boud k4e log k k by Tyomky [20] However if we look at Lemma 22 attetively, there is still some room to improve it Recall our discussio of Erdős cojecture i the itroductio: if the cojecture is true i geeral, the i order to miimize the umber of edges i a -hypergraph of a give matchig umber νh = /k, the hypergraph must be either a clique of size /k or the complemet of a clique of size /k { 1 1/k eh mi, 1 /k } k I this case, the umber of edges missig from H must be at least 1 2, which is far larger tha the boud 1 1 k+1 i Lemma 22 If i our proof of Theorem 23, the coefficiet before 1 ca be 1 chaged to a costat istead of the origial k+1, the theorem ca also be sharpeed to Ωk2 Based o this idea, i the rest of this sectio we are goig to prove Lemma 33, which asserts that eh c 1 for some costat c idepedet of ad k, ad ca be regarded as a stregtheig of Lemma 22 The we use it to prove our mai result of this paper, Theorem 35 I order to improve Lemma 22, istead of usig the usual matchig umber νh, it suffices to cosider its fractioal relaatio, which is defied as follows ν H = ma we w : EH [0, 1] e EH 1 subject to i e we 1 for every verte i 7 8 6

7 Note that ν H is always greater or equal tha νh O the other had, for our hypergraph we ca prove the same upper boud for the fractioal matchig umber ν H as i Lemma 21 Recall that H is the -uiform hypergraph o the 1 vertices {2,, }, whose edges are those -tuples i 1,, i correspodig to egative k-sums 1 + i1 + + ik < 0 Lemma 31 The fractioal matchig umber ν H is at most /k Proof Choose a weight fuctio w : EH [0, 1] which optimizes the liear program 8 ad gives the fractioal matchig umber ν H, the ν H = e EH we Two observatios ca be easily made: i if e EH, the i e i < 1 ; ii i 1 for ay i = 2,, sice { i } are i descedig order Therefore we ca boud the fractioal matchig umber i a few steps = 1 + we i + 1 we i i e i=2 1 + e EH 1 + e EH i e i=2 i we + i e = 1 ν H 1 + i=2 we ν H 1 we 1 i e e EH i e 1 we 1 = kν H 1 9 Lemma 31 follows from this iequality ad our assumptio that = 0 ad 1 > 0 The determiatio of the fractioal matchig umber is actually a liear programmig problem Therefore we ca cosider its dual problem, which gives the fractioal coverig umber τ H τ H = mi vi v : V H [0, 1] i subject to vi 1 for every edge e i e 10 By duality we have τ H = ν H /k Gettig a upper boud for eh is equivalet to fidig a fuctio v : V H [0, 1] satisfyig i V H vi /k that maimizes the umber of -tuples e where i e vi 1 Sice this umber is mootoe icreasig i every vi, we ca assume that it is maimized by a fuctio v with i V H vi = /k The followig lemma was established by Feige [9], ad later improved by He, Zhag, ad Zhag [11] It bouds the tail probability of the sum of idepedet oegative radom variables with give epectatios It is stroger tha Markov s iequality i the sese that the umber of variables m does ot appear i the boud Lemma 32 Give m idepedet oegative radom variables X 1,, X m, each of epectatio at most 1, the m { P r X i < m + δ mi δ/1 + δ, 1 } i=1 7

8 Now we ca show that the complemet of the hypergraph H has at least costat edge desity, which implies as a corollary that a costat proportio of the k-sums ivolvig 1 must be oegative Lemma 33 If Ck 2 with C 1, ad H is a -uiform hypergraph o 1 vertices with 1 fractioal coverig umber τ H = /k, the there are at least sets 2C! which are ot edges i H Proof Choose a weight fuctio v : V H [0, 1] which optimizes the liear programmig problem 10 Defie a sequece of idepedet ad idetically distributed radom variables X 1,, X, such that for ay 1 j, 2 i, X j = vi with probability 1/ 1 It is easy to compute the epectatio of X i, which is EX i = 1 1 vi = i=2 k 1 12 Now we ca estimate the umber of -tuples with sum less tha 1 The probability of the evet { i=1 X i < 1 } is basically the same as the probability that a radom -tuple has sum less tha 1, ecept that two radom variables X i ad X j might share a weight from the same verte, which is ot allowed for formig a edge However, we assumed that is much larger tha k, so this error term is ideed egligible for our applicatio Note that for i 1 < < i, the probability that X j = vi j for every 1 j is equal to 1/ 1 eh = = { i 1 < < i : vi vi < 1 } 1! 1! 1! 1! [ P r X 1 = vi 1,, X = vi ; distict i 1,,i i=1 [ P r X i < 1 ] P r X i = X j = vi l i=1 l i j [ P r [ P r i=1 i=1 X i < X i < 1 1 ] 2C ] ] X i < 1 13 The last iequality is because Ck 2 ad 1 3Ck + 2C for C 1, k 1, ad the sum of k 2 these two iequalities implies that C Defie Y i = X i k 1/ to ormalize the epectatios to EY i = 1 Y i s are oegative because each verte receives a oegative weight i the liear program 10 Applyig Lemma 32 8

9 to Y 1,, Y ad settig m =, δ = k/, we get P r X X < 1 = P r Y Y < k 1/ { k mi 2 k, 1 } 13 Whe > Ck 2 ad k 2, C 1, we have k 2 k > Ck2 k 2Ck 2 k = C 2C 1 13 Combiig 13 ad 14 we immediately obtai Lemma Corollary 34 If Ck 2 with C 1, the there are at least k-sums ivolvig 1 Now we are ready to prove our mai theorem: C! oegative Theorem 35 If 33k 2, the for ay real umbers 1,, with i=1 i 0, the umber of oegative k-sums is at least 1 Proof By the previous discussio, we kow that there are at least 2k oegative k-sums 1 usig oly 2,, By Corollary 34, there are at least ! 33! oegative k-sums ivolvig 1 I order to prove the theorem, we oly eed to show that for 33k 2, 2 1 2k ! 1 k + 2 Defie a ifiitely differetiable fuctio g = = It is ot! difficult to see g > 0 whe > Therefore 1 2k = g 1 g 2k [ 1 2k]g 1 = 2g 1 17 g = 1 2 k + 2 +! 1 k + 3! k 2! 2 k + 2! + + Combiig 17 ad 18, 1 2k 2g 1k 2 1 2! The last iequality follows from our assumptio 33k 2 1 k + 3! !

10 4 Hilto-Miler type results I this sectio we prove two Hilto-Miler type results about the miimum umber of oegative k-sums The first theorem asserts that for sufficietly large, if i=1 i 0 ad o i is large, the there are at least oegative k-sums Proof of Theorem 13 We agai assume that 1 ad i=1 i is zero Sice 1 is ot large, we kow that there eists a k 1-subset S 1 ot cotaiig 1, such that 1 + i S 1 i < 0 Suppose t is the largest iteger so that there are t subsets S 1,, S t, such that for ay 1 j t, S j is disjoit from {1,, j}, has size at most j ad j + i S j i < 0 As we eplaied above t 1 ad sice 1 we may also assume that S j cosists of the last S j idices i {1,, } By Corollary 34, for sufficietly large, there are at least oegative k-sums usig 1 Note also that after deletig 1 ad { i } i S1, the sum of the remaiig 1 S 1 k umbers is oegative Therefore, agai by Corollary 34, there are at least 1 14 oegative k-sums usig 2 but ot 1 I the et step, we delete 1, 2 ad { i } i S2 ad boud the umber of oegative k-sums ivolvig 3 but either 1 or 2 by Repeatig this process for 30 steps, we obtai N 1 [ ] > > 2 > where here we used the fact that > 2 ad is sufficietly large as a fuctio of k If 2 t < 30, by the maimality of t, we kow that the sum of t+1 with ay k 1 umbers with idices ot i {1,, t + 1} S t is oegative This gives us t+1 S t t oegative k-sums We ca also replace t+1 by ay i where 1 i t ad the ew k-sum is still oegative sice i t+1 Therefore, t 29 1 N t + 1 t + 1 > 2 for sufficietly large Thus the oly remaiig case is t = 1 Recall that 1 is ot large, ad hece 1 + k < 0 Suppose I is a -subset of [2, ] such that 1 + i I i < 0 If 2 I, the i I\{2} i < 0, this cotradicts the assumptio t = 1 sice I\{2} = k 2 2 Hece we ca assume that all the -subsets I 1,, I m correspodig to egative k-sums ivolvig 1 belog to the iterval [3, ] Let N 1 be the umber of oegative k-sums ivolvig 1, ad let N 2 be the umber of oegative k-sums usig 2 but ot 1, the [ ] 1 N N 1 + N 2 = m + N 2 10

11 I order to prove N 1 + 1, we oly eed to establish the followig iequality N 2 + m 1 20 Observe that the subsets I 1,, I m satisfy some additioal properties First of all, if two sets I i ad I j are disjoit, the by defiitio, 1 + l I i l < 0 ad 2 + l I j l 1 + l I j l < 0, summig them up gives l I i I j l < 0 with I i I j = 2, which agai cotradicts the assumptio t = 1 Therefore we might assume that {I i } 1 i m is a itersectig family By the Erdős-Ko-Rado theorem, m 2 1 = 1 3 k 2 The secod observatio is that if a -subset I [3, ] is disjoit from some I i, the 2 + i I i 0 Otherwise if 2 + i I i < 0 ad 1 + k I i k < 0, for the same reaso this cotradicts t = 1 Hece N 2 is bouded from below by the umber of -subsets I [3, ] such that I is disjoit from at least oe of I 1,, I m Equivaletly we eed to cout the distict -subsets cotaied i some J i = [3, ]\I i, all of which have sizes By the real versio of the Kruskal-Katoa theorem see E1331b i [13], if m = for some positive real umber, the N 2 O the other had, it is already kow that 1 m 3 k 2 = 3, thus 3 The oly remaiig step is to verify the followig iequality for i this rage, + Let f = [ f + 1 f =, ote that whe 4 = k 2 + k 2, ] [ = 0 k 2 k The last iequality is because whe is large, k 1 > 2k 2 Moreover, t is a icreasig fuctio for 0 < t < /2, so whe 4, k 2 = k 2 k 2 Therefore we oly eed to verify 21 for < k, which correspods to 1 m For m = 1, 21 is obvious, so it suffices to look at the case m 2 The umber of distict -subsets of J 1 or J 2 is miimized whe J 1 J 2 = k 2, which, by the iclusio-eclusio priciple, gives k 2 k 2 N 2 2 = + k 2 So 20 is also true for 2 m k 2 k It is easy to see that for k 3 ad sufficietly large, k 2 k For k = 2, we have = 3 ad 21 becomes , which is true ad completes the proof Remark 1 I order for all the iequalities to be correct, we oly eed > Ck 2 By carefully aalyzig the above computatios, oe ca check that C = 500 is eough 11 ]

12 Remark 2 Note that i the proof, the equality 20 holds i two differet cases The first case is whe m = 1, which meas 1 + k < 0 but ay other k-sums ivolvig 1 are oegative All the other oegative k-sums are formed by 2 together with ay -subsets ot cotaiig k+2,, This case is realizable by the followig costructio: 1 = k, 2 = 2, 3 = = k+1 = 1, k+2 = = = k+1 The secod case is i 21 whe = 4 ad = holds simultaeously, which gives k = 3 I this case, m = 3 4 = 3, ad the Kruskal-Katoa theorem holds with equality for the 4-subsets J 1,, J 3 That is to say, the egative 3-sums usig 1 are 1 + i + for 3 i 1, while the oegative 3-sums cotaiig 2 but ot 1 are 2 + i + j for 3 i < j 1 This case ca also be achieved by settig 1 = 2 = 1, 3 = = 1 = 1 2 3, ad = 3 2 For large, these are the oly two possible cofiguratios achievig equality i Theorem 13 Net we prove Theorem 14, which states that if i i 0 ad o i is moderately large, the at least a costat proportio of the k k-sums are oegative Proof of Theorem 14 Suppose t is the largest iteger so that there are t subsets S 1,, S t such that for ay 1 j t, S j is disjoit from {1,, j}, has at most j elemets, ad j + i S j i < 0 By the maimality of t, the sum of t+1 ad ay umbers i with idices from []\{1,, t + 1} S t is oegative, so there are at least t oegative k-sums usig t+1 Sice t+1 is ot 1 δ-moderately large, t 1 < 1 δ For sufficietly large, this is asymptotically equivalet to 1 tk < 1 δ Sice 1 tk > 1 we have t > k 2 δ tk, Recall that by Corollary 34, for each i = 1,, δ, k 2 i gives at least 1 i 1 14 oegative k-sums, therefore N 1 [ 1 14 δ δ 14k 2 k 2 δ = 1 δ/k 14k k 1 δ δ 1 14k k k k 2 δ ] 1 δ/k k + 1

13 Sice δ < 1, whe 1 δ, we have δ k + 1 δ Therefore settig gδ, k = δ1 δ 14k completes the proof 5 Cocludig remarks I this paper, we have proved that if > 33k 2, ay real umbers with a oegative sum have at least 1 oegative k-sums, thereby verifyig the Maickam-Miklós-Sighi cojecture i this rage Because of the iequality 2k k + C 1 1 we used, our method will ot give a better rage tha the quadratic oe, ad we did ot try hard to compute the best costat i the quadratic boud It would be iterestig to decide if the Maickam-Miklós-Sighi cojecture ca be verified for a liear rage > ck Perhaps some algebraic methods or structural aalysis of the etremal cofiguratios will help Feige [9] cojectures that the costat 1/13 i Lemma 32 ca be improved to 1/e This is a special case of a more geeral questio suggested by Samuels [18] He asked to determie, for a fied m, the ifimum of P rx X k < m, where the ifimum is take over all possible collectios of k idepedet oegative radom variables X 1,, X k with give epectatios µ 1,, µ k For k = 1 the aswer is give by Markov s iequality Samuels [18, 19] solved this questio for k 4, but for all k 5 his problem is still completely ope Aother itriguig objective is to prove the cojecture by Erdős which states that the maimum umber of edges i a r-uiform hypergraph H o vertices with matchig umber νh is eactly { } r[νh + 1] 1 νh ma, r r r The first umber correspods to a clique ad the secod case is the complemet of a clique Whe νh = 1, this cojecture is eactly the Erdős-Ko-Rado theorem [8] Erdős also verified it for > c r νh where c r is a costat depedig o r Recall that i our graph H we have νh /k ad r here is equal to, so if Erdős cojecture is true i geeral, we ca give a direct proof of costat edge desity i the complemet of H I this way we ca avoid usig fractioal matchigs i our proof But eve without the applicatio here, this cojecture is iterestig i its ow right The fractioal versio of Erdős cojecture is also very iterestig I its asymptotic form it says that if H is a r-uiform -verte hypergraph with fractioal matchig umber ν H =, where 0 < 1/r, the eh 1 + o1 ma { r r, 1 1 r} 22 r As poited out to us by Adrzej Ruciński, part of our reasoig i Sectio 3 implies that the fuctio A, k defied i the first page is precisely k mius the maimum possible umber of edges i a k-uiform hypergraph o vertices with fractioal coverig umber strictly smaller tha /k Ideed, give reals 1,, with sum zero ad oly A, k oegative k-sums, we may assume that the absolute value of each i is smaller tha 1/k otherwise simply multiply all of them by a sufficietly small positive real Net, add a sufficietly small positive ɛ to each i, keepig 13

14 each i smaller tha 1/k ad keepig the sum of ay egative k-tuple below zero this is clearly possible Note that the sum of these ew reals, call them i, is strictly positive ad the umber of positive k-sums is A, k Put νi = 1/k i ad observe that i νi < /k ad the k-uiform hypergraph whose edges are all k-sets e for which i e νi 1 has eactly k A, k edges Therefore, there is a k-uiform hypergraph o vertices with fractioal coverig umber strictly smaller tha /k ad at least k A, k edges Coversely, give a k-uiform hypergraph H o vertices ad a fractioal coverig of it ν : V H [0, 1] with i νi = /k δ < /k ad i e νi 1 for each e EH, oe ca defie i = 1 k δ νi to get a set of reals whose sum is zero, i which the umber of oegative k-sums is at most k EH as the sum of the umbers i for every k-set formig a edge of H is at most 1 kδ 1 < 0 This implies the desired equality, showig that the problem of determiig A, k is equivalet to that of fidig the maimum possible umber of edges of a k-uiform hypergraph o vertices with fractioal coverig umber strictly smaller tha /k Note that this is equivalet to the problem of settlig the fractioal versio of the cojecture of Erdős for the etremal case of fractioal matchig umber < /k Although the fractioal versio of Erdős cojecture is still widely ope i geeral, our techiques ca be used to make some progress o this problem Combiig the approach from Sectio 3 ad the above metioed results of Samuels we verified, i joit work with Frakl, Rödl ad Ruciński [1], cojecture 22 for certai rages of for 3 ad 4-uiform hypergraphs These results ca be used to study a Dirac-type questio of Dayki ad Häggkvist [6] ad of Há, Perso ad Schacht [10] about perfect matchigs i hypergraphs For a r-uiform hypergraph H ad for 1 d r, let δ d H deote the miimum umber of edges cotaiig a subset of d vertices of H, where the miimum is take over all such subsets I particular, δ 1 H is the miimum verte degree of H For r that divides, let m d r, deote the smallest umber, so that ay r-uiform hypergraph H o vertices with δ d H m d r, cotais a perfect matchig Similarly, let m d r, deote the smallest umber, so that ay r- uiform hypergraph H o vertices with δ d H m d r, cotais a perfect fractioal matchig Together with Frakl, Rödl ad Ruciński [1] we proved that for all d ad r, m d r, m d r, ad further reduced the problem of determiig the asymptotic behavior of these umbers to some special cases of cojecture 22 Usig this relatio we were able to determie m d r, asymptotically for several values of d ad r, which have ot bee kow before Moreover, our approach may lead to a solutio of the geeral case as well, see [1] for the details Ackowledgmet We would like to thak Adrzej Ruciński for helpful discussios ad commets, ad Nati Liial for ispirig coversatios ad itriguig questios which led us to the results i Sectio 4 Refereces [1] N Alo, P Frakl, H Huag, V Rödl, A Ruciński ad B Sudakov, Large matchigs i uiform hypergraphs ad the cojectures of Erdos ad Samuels, submitted 14

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