The minimum number of nonnegative edges in hypergraphs

Transcription

1 The minimum number of nonnegaive edges in hypergraphs Hao Huang DIMACS Rugers Universiy New Brunswic, USA Benny Sudaov Deparmen of Mahemaics ETH 8092 Zurich, Swizerland Submied: May 26, 2014; Acceped: Jul 3, 2014; Published: Jul 10, 2014 Mahemaics Subjec Classificaions: 05C35, 05C65, 05D05 Absrac An r-uniform n-verex hypergraph H is said o have he Manicam-Milós-Singhi MMS propery if for every assignmen of weighs o is verices wih nonnegaive sum, he number of edges whose oal weigh is nonnegaive is a leas he minimum degree of H In his paper we show ha for n > 10r 3, every r-uniform n-verex hypergraph wih eual codegrees has he MMS propery, and he bound on n is essenially igh up o a consan facor This resul has wo immediae corollaries Firs i shows ha every se of n > 10 3 real numbers wih nonnegaive sum has a leas n 1 1 nonnegaive -sums, verifying he Manicam-Milós-Singhi conjecure for his range More imporanly, i implies he vecor space Manicam- Milós-Singhi conjecure which saes ha for n 4 and any weighing on he 1-dimensional subspaces of F n wih nonnegaive sum, he number of nonnegaive -dimensional subspaces is a leas [ n 1 1 We also discuss wo addiional generalizaions, which can be regarded as analogues of he Erdős-Ko-Rado heorem on -inersecing families 1 Inroducion Given an r-uniform n-verex hypergraph H wih minimum degree δh, suppose every verex has a weigh w i such ha w w n 0 How many nonnegaive edges mus H have? An edge of H is nonnegaive if he sum of he weighs on is verices is 0 Research suppored in par by NSF gran DMS Research suppored in par by SNSF gran and by a USA-Israel BSF gran he elecronic journal of combinaorics , #P37 1

2 Le e + H be he number of such edges By assigning weigh n 1 o he verex wih minimum degree, and 1 o he remaining verices, i is easy o see ha he number of nonnegaive edges can be a mos δh I is a very naural uesion o deermine when his easy upper bound is igh, which leads us o he following definiion Definiion 11 A hypergraph H wih minimum degree δh has he MMS propery if for every weighing w : V H R saisfying x vh wx 0, he number of nonnegaive edges is a leas δh The uesion, which hypergraphs have MMS propery, was moivaed by wo old conjecures of Manicam, Milós, and Singhi [9, 10, boh of which were raised in heir sudy of so-called firs disribuion invarian of cerain associaion schemes Conjecure 12 Suppose n 4, and we have n real numbers w 1,, w n such ha w 1 + +w n 0, hen here are a leas n 1 1 subses A of size saisfying w i A w i 0 The second conjecure is an analogue of Conjecure 12 for vecor spaces Le V be a n- dimensional vecor space over a finie field F Denoe by [ V he family of -dimensional subspaces of V, and he -Gaussian binomial coefficien [ n is defined as n i 1 0 i< i 1 Conjecure [ 13 Suppose n 4, and V is he n-dimensional vecor space over F Le w : V 1 R be a weighing on he one-dimensional subspaces of V such ha v [ V 1 wv = 0, hen he number of -dimensional subspaces S wih v [ wv 0 V 1,v S is a leas [ n 1 1 Conjecure 12 can be regarded as an analogue of he famous Erdős-Ko-Rado heorem [4 The laer says ha for n 2, a family of -subses of [n wih he propery ha every wo subses have a nonempy inersecion has size a mos n 1 1 In boh problems, he exremal examples correspond o a sar, which consiss of subses conaining a paricular elemen in [n The Manicam-Milós-Singhi conjecure has been open for more han wo decades, and various parial resuls were proven There are several wors verifying he conjecure for small [6, 8, 11 Bu mos of he research focus on proving he conjecure for every n greaer han a given funcion f Manicam and Milós [9 verified he conjecure for n Laer Tyomyn [14 improved his bound o n e c log log Alon, Huang, and Sudaov [1 obained he firs polynomial bound n > 33 2 Laer, Franl [5 gave a shorer proof for a cubic range n A linear bound n was obained by Porovsiy [12 He reduced he conjecure o finding a -uniform hypergraph on n verices saisfying he MMS propery similar echniues were also employed earlier in [9 The second conjecure, Conjecure 13, was very recenly proved by Chowdhury, Saris, and Shahriari [3 simulaneously wih our wor They also proved a uadraic bound n 8 2 for Conjecure 12 We observe ha boh conjecures can be reduced o proving ha cerain hypergraph has he MMS propery For he firs conjecure, simply le he hypergraph H 1 be he he elecronic journal of combinaorics , #P37 2

3 complee [ -uniform hypergraph on n verices For he second conjecure, one can ae he -uniform hypergraph H 1 2 wih verex se [ V 1 and le edges correspond o -dimensional subspaces Boh hypergraphs are regular, and moreover he codegree of every pair of verices is he same I is emping o conjecure ha all such hypergraphs saisfy he MMS propery The reuiremen ha all he codegrees are eual may no be dropped For insance, he igh Hamilonian cycle he edges are consecuive r-uples modulo n when n 1 mod r is no MMS This can be seen by choosing he weighs wxr + 1 = n for x = 0,, n/r and all he oher weighs o be n+r, which resuls in only r 1 r 1 nonnegaive edges, as opposed o he fac ha he degree is r Our main heorem indeed confirms ha eual codegrees imply he MMS propery Theorem 14 Le H be an r-uniform n-verex hypergraph wih n > 10r 3 and all he codegrees eual o λ Then for every weighing w : V H R wih v w v 0, we have e + H δh Moreover in he case of eualiy, all nonnegaive edges form a sar, ie, conain a fixed verex of H The lower bound on n in his heorem is igh up o a consan facor Our resul immediaely implies wo corollaries Firs i verifies Conjecure 12 for a weaer range n Ω 3 Moreover i also provides a proof of Conjecure 13 As menioned earlier, here are some suble connecions beween Manicam-Milós- Singhi conjecure and he Erdős-Ko-Rado heorem on inersecing families In [4, Erdős, Ko and Rado also iniiaed he sudy of -inersecing families any wo subses have a leas common elemens They show ha for <, here exiss an ineger n 0, such ha for all n n 0, he larges -inersecing family of -ses are he -sars, which are of size This resul is euivalen o saying ha in he -uniform hypergraph H whose verices are -subses of [n and edges correspond o -subses of [n, he maximum inersecing sub-hypergraph has size The following heorem says ha for large n, his hypergraph has he MMS propery Noe ha his is no implied by Theorem 14, because he codegree of wo verices as -subses depends on he size of heir inersecion Theorem 15 Le, be posiive inegers wih >, n > C 3+3 for sufficienly large C and le { } X [n be a weigh assignmen wih X [n 0 Then here are always a leas subses T of size such ha X T 0 This resul can be regarded as an analogue of he -inersecing version of he Erdős- Ko-Rado heorem Moreover, he Manicam-Milós-Singhi conjecure is a special case of his heorem corresponding o = 1 and = r Using a similar proof one can also obain a generalizaion of he vecor space version of Manicam-Milós-Singhi conjecure Theorem 16 Le, be posiive inegers wih >, n > C for sufficienly large C, V be he n-dimensional vecor space over F and le { } X [ V be a weigh assignmen wih X [ V 0 Then here are always a leas [ -dimensional subspaces T such ha X [ V,X T 0 he elecronic journal of combinaorics , #P37 3

4 The res of he paper is organized as follows In Secion 2 we prove Theorem 14 and deduce Conjecure 13 as a corollary In Secion 3 we will have wo consrucions, showing ha he n > Ω 3 bound for Theorem 14 is essenially igh The ideas presened in Secion 2 are no enough o prove Theorem 15 Hence, in Secion 4 we develop more sophisicaed echniues o prove his heorem We also sech he lemmas needed o obain Theorem 16 and leave he proof deails o he appendix The final secion conains some open problems and furher research direcions 2 Eual codegrees and MMS propery In his secion we prove Theorem 14 Wihou loss of generaliy, we may assume ha V H = [n, and he weighs are 1 = w 1 w 2 w n, such ha n i=1 w i = 0 Throughou his secion w i is also used o indicae he verex i when here is no confusion Suppose he number of edges in H is e By double couning, we have ha H is d-regular wih d = n 1λ and ha dn = re By considering he 2r-h larges weigh w r 1 2r, we will verify Theorem 14 for he following hree cases respecively: i w 2r 1 ; ii w 2r 2 2r 1 ; 2r 1 and iii w 2r 2 2r 1 2r Lemma 21 If w 2r 1, hen e + H d 2r 2 Proof Firs we show ha among he d edges conaining w 1, he number of negaive edges is a mos 5d Denoe hese negaive edges by e 6r 1,, e m and he nonnegaive edges by e m+1,, e d By he definiion of a negaive edge, for every 1 i m we have w j < w 1 = 1 j e i \{1} Summing hese ineualiies, we ge m Now we consider he sum d i=m+1 i=1 j e i \{1} w j < m j e i \{1} w j and rewrie i as j α jw j The sum of coefficiens α j s is eual o d mr 1 Noe ha in his sum w 2,, w 2r each appears a mos λ imes heir codegree wih {1} and hey are bounded by 1, so in oal hey conribue no more han 2rλ The remaining variables w 2r+1,, w n conribue less han d mr 1w 2r < d m 2r Combining hese hree esimaes, we obain ha w j < m + 2rλ + d m 2r 1 e j e\{1} By double couning, he lef hand side is eual o λw w n = λ Comparing hese wo uaniies and doing simple calculaions we ge m < 2rλ + d 2r + 1 < d 5r + d 2r + 1 < 5d 6r he elecronic journal of combinaorics , #P37 4

5 Here we used ha n > 10r 3 and λ = r 1 d < n 1 d/10r2 Therefore we may assume here are a leas 1 5 d nonnegaive edges hrough w 6r 1 If every edge hrough w 1 is posiive, hen we are already done; so we assume ha here exiss a negaive edge e hrough w 1 Suppose w u is he larges posiive weigh of verex no conained in e Such u exiss since oherwise n i=1 w i < 0, so we may assume ha u 2 and {1,, u 1} e We claim ha in his case here are many nonnegaive edges hrough w u which are disjoin from e Consider he se S of r-uples consising of all he edges hrough w u which are disjoin from e Since each of he r verices of e has a mos λ common neighbors wih w u, we have S d rλ Denoe by S he se of negaive edges in S and consider he sum f S j f\{u} w j Obviously i is a mos w u S Rewrie his sum as λ j e {u} α jw j Since all codegrees are λ, α j [0, 1 and j e {u} α j = r 1 S /λ, which implies ha j e {u} 1 α j = n r 1 r 1 S /λ Therefore w u < j e {u} w j = j e {u} α j w j + j e {u} 1 α j w j < S λ w u + n r 1 r 1 S /λ w u, The firs ineualiy uses ha he sum of all he weighs is zero and ha e is a negaive edge, so j e w j < 0 To see he second ineualiy, jus observe ha w j w u for every j e {u} By simplifying he las ineualiy we ge S < n r Therefore he number of nonnegaive edges conaining w 2 ha are disjoin from e is a λ r leas S S > d rλ n r r λ = n r3 2r 2 + 2r d, rn 1 which is greaer han 5d 6r if n > 10r3 These nonnegaive edges, ogeher wih he 1 5 6r d nonnegaive edges hrough w 1, already give more han d nonnegaive edges Lemma 22 If w 2r 1 2r, hen e+ H d Proof Firs we claim for any 1 i 2r, here are a leas 3 d nonnegaive edges 5r conaining w i Le S i be he se of negaive edges conaining w i, hen for any edge e S i, j e\{i} w j < w i Summing up hese ineualiies, we have e S i j e\{i} w j < S i w i Lie he previous case, suppose he lef hand side can be rewrien as λ j i α jw j, hen α j [0, 1, and j i α j = r 1 S i /λ, which implies j i 1 α j = n 1 r 1 S i /λ he elecronic journal of combinaorics , #P37 5

6 Since j i w j = w i, we have w i = j i α j w j + j i 1 α j w j < S i λ w i + 1 j 2r S i λ w i + 2rw 1 + Subsiuing λ = r 1 n 1 d and w i w 2r 1 2r, gives w j + w i 1 α j j>2r n 1 r 1 S i λ S i n + 4r 2 λ r = r 1n + 4r2 d, rn 1 w i which is less han 1 3 d when n > 5r 10r3 So here are a leas 3 d nonnegaive edges 5r conaining w i Noe ha for 1 i < j 2r, w i and w j are simulaneously conained in a mos λ edges Therefore he oal number of nonnegaive edges is a leas 2r 3 5r d 2r 2 λ 6 5 d 2r2 r 1 d n 1 When n > 10r 3, his gives more han d nonnegaive edges Lemma 23 If 1 2r 2 w 2r 1 2r, hen e+ H d Proof Le be he index such ha w 2rw 2r and w +1 < 2rw 2r Since w 1 = 1 2rw 2r such exiss and is beween 1 and 2r For arbirary 1 i, le T i be he se of negaive edges conaining w i Similarly as before, we assume w j = λ α j w j e T i j e\{i} Then j i α jw j < w i T i /λ, and j i 1 α j = n 1 r 1 T i /λ We also have j i w i = w j = α j w j + 1 α j w j j i j i j i < T i λ w i + 1 α j w j + α j w j + 1 j <j 2r1 1 α j w j j>2r T i λ w i + + 2r w + n 1 r 1 T i /λw 2r T i λ w i + + 2r w + n 1 r 1 T i /λ w 2r Suppose T i /λ 1 Since w i w, we hen have Ti λ 1 2r n 1 r 1 T i /λ w 2r 2 w 2r rw 2r he elecronic journal of combinaorics , #P37 6

7 Using ha n > 10r 3, 2r and λ = r 1 d and rearranging he las ineualiy gives n 1 T i 2r n 1 + r 1 + 2r + 1 3r 1 2r λ 2r n 1 r 1 + 2r r 1 2r n 1 d < 7 15 d Since λ < d/10r 2, he above ineualiy also holds when T i < λ Therefore here are a leas 8 d nonnegaive edges hrough w 15 i This complees he proof for 2, as he number of nonnegaive edges hrough w 1 and w 2 is already a leas 16d λ > d when 15 n > 10r 3 If = 1, i means ha w 2 < 2rw 2r For 2 i 2r, as before denoe by U i he se of all he negaive edges hrough w i Then similarly we define α j w j f U i j f\{i} w j = λ j i Noe ha α i [0, 1, j i α jw j < w i U i /λ and j i 1 α j = n 1 r 1 U i /λ So We have w i = j i w j = j i α j w j + j i 1 α j w j U i λ w i + U i λ w i rw 2 + n 1 r 1 U i /λw i 2r i=1 w i + j>2r 1 α j w j r U i λ n w i 1 + 2rw 2 2r 2 w 2r + 2r 2rw 2r = 6r 2 w 2r 6r 2 w i Therefore when n > 10r 3, U i 6r2 + n r r 1 1 n 1 d 2 d 5r Hence here are a leas 2 d nonnegaive edges hrough every w 5r i when 2 i 2r, ogeher wih he 8 d nonnegaive edges hrough w 15 1 Thus he oal number of nonnegaive edges is a leas 8 15 d + 2 2r 4 d2r 1 λ = 5r r 2r3 3r 2 + r d > d, n 1 where we used ha λ = r 1 n 1 d and n > 10r3 Combining Lemma 21, Lemma 22 and Lemma 23 we show ha e + H d From he proofs i is no hard o see ha when n > 10r 3 he only way o achieve he ineualiy is when he nonnegaive edges form a sar, ie conain a fixed verex of H This concludes he proof of Theorem 14 he elecronic journal of combinaorics , #P37 7

8 Nex we use Theorem 14 o prove he vecor space analogue of Manicam-Milós- Singhi conjecure Proof of Conjecure 13: Le H be he hypergraph such ha he verex se V H consiss of all he 1-dimensional subspaces of V = F n Obviously he number of verices is eual o [ n Every -dimensional subspace of V defines an edge of H which conains 1 exacly [ 1 verices Therefore H is an r-uniform hypergraph on n verices wih r = [ 1 and n = [ n Since every wo 1-dimensional subspaces span a uniue 2-dimensional 1 subspace, he codegree of any wo verices in H is eual o [ n 2 Applying Theorem 14, 2 as long as n > 10r 3, he minimum number of nonnegaive edges in H is a leas eual o is degree, which is eual o [ n 1 Acually he condiion ha 1 n > 10r 3 is euivalen o n > Since n 4, we have n Moreover 4 1/ 1 3 = / From 2, we also have > 10 if 3 Therefore n > For = 2, i is no hard o verify ha he ineualiy is sill saisfied when 3 The only remaining case is when, = 2, 2 Again i is easy o chec ha he ineualiy holds when n 9 The case n = 8 was resolved by Manicam and Singhi [10, who proved heir conjecure when divides n Remar The saemen of Conjecure 13 is nown o be false only for n < 2 Hence, i would be ineresing o deermine he minimal n = n which implies his conjecure Noe ha Theorem 14 can be used o prove he asserion of Conjecure 13 also for n < 4 For example i shows ha his conjecure holds for n 3 and 5, n and 3 or n and all For large he proof will wor already saring wih n = The ighness of Theorem 14 In he previous secion, we show ha for every r-uniform n-verex hypergraph wih eual codegrees and n > 10r 3, he minimum number of nonnegaive edges is always achieved by he sars Here we discuss he ighness of his resul As a warm-up example, recall ha a finie projecive plane has N 2 + N + 1 poins and N 2 + N + 1 lines such ha every line conains N + 1 poins Moreover every wo poins deermine a uniue line, and every wo lines inersec a a uniue poin If we regard poins as verices and lines as edges, his naurally corresponds o a N + 1-uniform N + 1-regular hypergraph wih all codegrees eual o 1 Le us assign weighs 1 o he N + 1 poins on a fixed line l, and weighs he elecronic journal of combinaorics , #P37 8

9 N+1 o he oher poins Obviously he sum is nonnegaive On he oher hand every N 2 line oher han l conains a mos one poin wih posiive weigh, hus is sum of weigh is a mos 1 N N+1 < 0 Therefore here is only one nonnegaive edge This already N 2 gives us a hypergraph wih n r 2 ha is no MMS The nex heorem provides an example of a hypergraph wih n r 3 for which here is a configuraion of edges, differen from a sar, ha also achieves he minimum number of nonnegaive edges Theorem 31 For infiniely many r here is an r-uniform hypergraph H wih eual codegrees on r 3 2r 2 + 2r verices, and a weighing w : V H R wih nonnegaive sum, such ha here are δh nonnegaive edges ha do no form a sar Proof Le r = + 1, where is a prime power Denoe by F he finie field wih elemens Define a hypergraph H wih he verex se V H consising of poins from he 3-dimensional projecive space P G3, F Here P Gn, F = F n+1 \{0}/, wih he euivalence relaion x 0,, x n σx 0,, σx n, where σ is an arbirary number from F I is easy o see ha n = V H = = r 3 2r 2 +2r Every 1-dimensional subspace of P G3, F defines an edge of H wih + 1 = r elemens I is no hard o chec ha H is d-regular for d = , and every pair of verices has codegree 1 Now we assign he weighs o V H in he following way Le S be he se of poins of a 2-dimensional projecive subspace of V H, hen S = Every verex from S receives weigh 1, and every verex ouside S has weigh 2 ++1, such ha he oal 3 weigh is zero Noe ha every edge has size + 1, so if i conains a mos one verex from S, is oal weigh is a mos < 0 Therefore every nonnegaive edge 3 mus conain a leas wo verices from S Since S is a subspace, he lines conaining 2 poins from S are compleely conained in S There are precisely = d lines in S hese are all he nonnegaive edges in H and hey do no form a sar Finally, we give an example which shows ha one migh find hypergraphs wih n r 3 and weighs such ha he number of nonnegaive edges is sricly smaller han he verex degree Recall ha in number heory, a Mersenne prime is a prime number of he form 2 n 1 Theorem 32 If and + 1 are boh prime powers, hen here exiss a + 1-uniform regular hypergraph H on verices wih all codegrees eual o 1, and an assignmen of weighs wih nonnegaive sum such ha here are sricly less han nonnegaive edges in H In paricular if here are infiniely many Mersenne primes, hen we obain infiniely many such hypergraphs Proof Le V H = V 1 V 2, such ha V 1 = , and V 2 = We firs ae H 1 o be he projecive plane P G2, F on V 1 wih edges corresponding o he projecive lines In oher words H 1 is a + 1-uniform hypergraph wih degree + 1 and codegree 1 The hypergraph H 2 consiss of some + 1-uples ha inersec V 1 in exacly one he elecronic journal of combinaorics , #P37 9

10 verex and inersec V 2 in verices, such ha eh 2 = H 3 is a + 1-uniform hypergraph on V 2 wih 3 edges We will carefully define he edges of H 2 and H 3 soon We hope H 2 and H 3 o saisfy he following properies: i for every pair of verices u V 1 and v V 2, heir codegree in H 2 is eual o 1; ii noe ha every edge in H 2 naurally induces a cliue of size in V 2 2 ; while every edge in H3 induces a cliue of size + 1 in V 2 2 We hope hese cliues form an edge pariion of he complee graph K V2 = K 2 +1 I is no hard o see ha if i, ii are boh saisfied, hen he hypergraph H = H 1 H 2 H 3 has codegree δ = 1 Noe ha H is a regular hypergraph wih degrees eual o δ n 1 r 1 = = Now we assign weighs o V H, such ha every verex in V 2 receives a weigh 1, while every verex in V 1 receives a weigh 2 +1, so he oal weigh is zero If an edge is nonnegaive, i mus conain a leas wo verices from V 1, since < Such an edge can only come from H 1 However we have eh 1 = , which is sricly smaller han he degree Therefore wha remains is o show he exisence of H 2 and H 3 saisfying i, ii In oher words, we need o find a cliue pariion in ii wih K 2 +1 = 3 K K Moreover, condiion i reuires ha he family of K s can be pariioned ino K -facors A naural idea is o pariion [ = S 1 S wih S i = + 1 Observe ha he projecive plane P G2, F defines a cliue pariion K = K +1 By removing one verex from i, we obain a pariion K 2 + = + 1 K 2 K +1 By doing his for every S i, we ge he 3 copies of K +1 we wan, and + 1 copies of K, which clearly form a K -facor, since he + 1 copies of K from each S i are pairwise disjoin We sill need o find an edge pariion of he balanced complee -parie graph K 2 +,, 2 + ino copies of K, so ha hey also can be grouped ino 2 + disjoin K -facors Suppose we now ha + 1 is also a prime power Label he verices in K 2 +,, 2 + by x, y, z where x F, y F, and z F +1 Two verices x, y, z and x, y, z are adjacen iff x x Now we define cliues C i,j,,l s for i, F and j, l F +1 The cliue C i,j,,l consiss of verices in he form of x, i + x, j + lfx for all x F, where f is a fixed injecive map from F o F +1 Suppose x, y, z and x, y, z wih x x are boh conained in he cliue C i,j,,l, hen we have i + x = y i + x = y j + lfx = z j + lfx = z Since x x, he firs wo euaions uniuely deermine i and Moreover, fx and fx are differen elemens of F +1 since f is injecive, hus j, l are also uniuely deermined he elecronic journal of combinaorics , #P37 10

11 Therefore {C i,j,,l } forms a K -pariion of he edges of K 2 +,, 2 +, and i is no hard o see ha hey can be pariioned ino K -facors by fixing and l By he above discussions, if we have boh of and + 1 o be powers of prime, in paricular when = 2 n 1 is a Mersenne prime, hen we can explicily consruc he hypergraph 4 Two addiional generalizaions In he nex wo subsecions we discuss generalizaions of he wo Manicam-Milós-Singhi conjecures and prove Theorem 15 and Theorem 16 For = 1 hey follow from Theorem 14, hus we can assume ha 2 In ha case, as we menioned earlier in he inroducion, hese wo heorems are no direc conseuences of Theorem 14 because he codegrees in he corresponding hypergraphs are no eual 41 Generalizaion of MMS In his subsecion we will prove Theorem 15 This reuires some new ideas and echniues since direc adapaion of he proof of Theorem 14 does no wor Indeed, i is easy o consruc a weighing such ha here is no nonnegaive edge hrough he verex a -se of maximal weigh For example say = 2, one can ae w {1,2} = 1, he weighs of all he 2n 4 pairs conaining 1 or 2 o be n 3, and he res o have weighs roughly 2 For 10 5 sufficienly large n, no -se conaining {1, 2} has nonnegaive oal weighs Firs we prove a simple lemma from linear algebra Lemma 41 Suppose he s s lower riangular marix β = {β i,j } saisfies ha β i,i > 0 and for every j < i, 0 β i,j β i, Then for a given vecor b = b 1,, b s such ha b 1 b s 0, he euaion b = γ β has a uniue soluion γ = γ 1,, γ s and moreover 0 γ i b i /β i,i Proof The exisence and uniueness of γ follow from he fac ha β is inverible Nex we inducively prove 0 γ i b i /β i,i We sar from γ s, from he euaion we now b s = γ s β s,s So γ s = b s /β s,s and he inducive hypohesis is rue Suppose 0 γ i b i /β i,i for every i > Now from he linear euaion, we have b = β, γ +β +1, γ +1 +β s, γ s Since γ i and β i, are nonnegaive for i >, we have γ b /β, Noe ha β i,j is increasing in j, so for every + 1 i s, β i, β i,+1 Therefore b β, γ + s i=+1 β i,+1γ i = β, γ + b +1 Since 0 b +1 b, we now ha γ 0 Wihou loss of generaliy, we may assume ha X [n = 0; and w {1,,}, or alernaively wrien as w [, has he larges posiive weigh We may le w [ = 1, hen 1 for every -se X Throughou his secion, we also assume ha n > C 3+3, here C is some sufficienly large consan The nex lemma shows ha if he sum of weighs of cerain edges is very negaive, hen we already have enough nonnegaive edges he elecronic journal of combinaorics , #P37 11

12 Lemma 42 If for some subse L =, X L =0 L Y, Y = L X Y n, and X L 1, hen here are more han nonnegaive edges in H Proof We may rewrie he lef hand side of he ineualiy as n 2 n n = n 1 1 X L 1 Here we le b j = n 2+j 2+j n > C 3+3, his implies 2 b j j=0 X L =j 2 b j j= X L =1 X L =j X L = 1 Noe ha X L 1 = X L 1 Since n n n 1 For a fixed ineger 0 y 1, denoe by D y he number of nonnegaive -ses Z wih Z L = y If D y > hen we are done Oherwise assume Dy for every y We esimae he following sum: Z L =y, Z = X Z Since every nonnegaive -se conribues o he sum a mos, i is a mos Dy By double couning, he above sum also euals y j=0 β y,j X L =j, where β y,j = j n 2+j y j y+j, noe ha βy,j = 0 when j < + y When j + y, since n, for fixed y, β y,j is increasing in j Also noe ha b j is decreasing in j Le γ = γ 0,, γ 2 be he uniue soluion of he sysem of euaions b = γ β, hen by Lemma 41 2 b j j=0 X L =j Since b y /β y,y = 2 b j j=0 1 1 X L =j = 2 2 β y,j γ y j=0 y=j n 2+y 2+y n For n > C 3+3 his conradics 1 X L =j 2 γ y y=0 / n 2+y n 2 = γ y y=0 n / n n 2 y j=0 2 β y,j y=0 b y β y,y We have +1 n X L =j n he elecronic journal of combinaorics , #P37 12

13 We now assume ha he -h larges weigh in H is w P and consider several cases Lemma 43 If w P > 1 2, here are more han nonnegaive edges in he hypergraph H Proof We will show ha every verex whose weigh is larger han w P is conained in 3 a leas 2 nonnegaive edges, oherwise here are already > nonnegaive edges For simpliciy we jus need o prove his saemen for w P iself Suppose here are S negaive edges conaining w P, which are denoed by e 1,, e S as -subses And e S+1,, e are he oher hus nonnnegaive edges conaining w P We have i=1 P X e i = S i=1 P X e i + w P S + w P i=s+1 P X e i n S 1 + n Here we used ha here are a mos ses X whose weigh is larger han w P bu always 1, and he number of imes every such se appears in he sum is a mos 1 1 If S 1 3 2, hen he above expression is a mos n w P 3 2 w P +1, n which can be furher bounded by n w P +1 n 1 n < 2! n 13 w 1 P 13 2 The las ineualiy uses ha n > C 3+3, > 2 and herefore ! = 1 Since we also have ! 12 X P = w P 1, Lemma 42 for L = P immediaely gives > nonnegaive edges Therefore we can assume ha for he ses wih larges weighs, he number of 3 nonnegaive edges conaining each such se is a leas 2 Using he union bound, he number of nonnegaive edges is a leas which is also larger han 3 n 2 2 n 1, 1 The nex lemma covers he case when w P is smaller han 1 2, and here are significan number of negaive edges conaining {1,, } he elecronic journal of combinaorics , #P37 13

14 Lemma 44 If he -h larges weigh w P is smaller han 1/ 2, and here are less han 1 1 nonnegaive edges conaining {1,, }, hen here are a leas nonnegaive edges in H Proof We consider all he -uples conaining {1,, }, similarly as before suppose here are S 1 negaive edges e1,, e S and nonnegaive edges e S+1,, e, we ge [ Z, Z = X Z,X [ = S i=1 X e i,x [ + i=s+1 X e i,x [ S + 1 n S 2 S 1 n S! 1 n 1 1 n The firs ineualiy is by bounding he larges weighs in he second sum by 1 and he res by 1 I also uses he fac ha wo ses are conained in a mos 1 1 edges 2 Since S 1 and 2, we have 1 n n [ Z, Z = X Z,X [ n n For large n he righ hand side is a mos 1 3 We also have X [ = w [ = 1 Now we once again can apply Lemma 42 for L = {1,, } o show he exisence of > nonnegaive edges I remains o prove he case when {1,, } is conained in a leas 1 1 nonnegaive edges Lemma 45 If {1,, } is conained in a leas 1 1 nonnegaive edges, hen here are a leas nonnegaive edges in H Proof Noe ha if every edge conaining {1,, } is nonnegaive, his already gives nonnegaive edges and he lemma is proved So we may assume ha here is a negaive edge f as -subse hrough {1,, } wih X f < 0 Suppose he larges weigh ouside he edge f is w Q, where Q f 1 Now we define new weighs w, such ha { if X f = /w Q oherwise he elecronic journal of combinaorics , #P37 14

15 Then for every X f, 1 and w Q = 1 Now we consider all he -uples conaining he -se Q As usual, assume ha S of hem has negaive sum according o w If S 1 3 2, we have he following esimae: n S + S Q Y, Y = X Y,X Q n n ! 1 n 12 Noe ha since X f < 0, we have X Q w X = X Q /w Q X f /w Q If we apply Lemma 42 for L = Q and he weigh = w X /2, we ge > nonnegaive edges for he new weigh funcion w Noe ha every such nonnegaive edge can no share wih f a common -subse, oherwise is oal weigh is a mos 1 < 0 Hence hese nonnegaive edges are also nonnegaive edges for he original weigh funcion w By he above discussion, i remains o consider he case S < Then here are a leas 3 2 nonnegaive edges conaining Q, ogeher wih he 1 1 nonnegaive edges conaining {1,, } Since {1,, } and w Q have codegree a mos 1 1 < 1 2, we have in oal more han nonnegaive edges 42 Generalizaion of vecor MMS Our echniues from he previous secion also allow us o prove a generalizaion of he vecor space version of Manicam-Milós-Singhi conjecure Since he proof of his resul is very similar o ha of Theorem 15 we only sae he appropriae varians of he lemmas involved The deailed proofs of hese lemmas can be found in he appendix of his paper The proof of Theorem 16 follows immediaely from combining hese lemmas As before, we define he hypergraph H o have he verex se [ V and every edge corresponds o a -dimensional subspace I is easy o chec ha he hypergraph is [ -uniform on [ n verices Lie he previous secion, we also assume ha [ is he -dimensional subspace wih w [ = 1 and for every X, 1 All he following lemmas are proven under he assumpion ha n > C for sufficienly large consan C he elecronic journal of combinaorics , #P37 15

16 Lemma 46 If for some -dimensional subspace L, L Y,Y [ V L X Y 1 24 [ 2 [ n, and X L 1, hen here are more han [ nonnegaive edges in H We now assume ha he 3 [ -h larges weigh in H is w P, and consider he following cases Lemma 47 If w P > 1/4 [ 2, here are more han [ nonnegaive edges in H Lemma 48 If w P 1/4 [ 2 1, and here are less han 1 [ 2[ nonnegaive edges conaining [, hen here are a leas [ nonnegaive edges in H Lemma 49 If [ is conained in a leas 1 2[ 1 [ nonnegaive edges, hen here are a leas [ nonnegaive edges in H 5 Concluding Remars A r n,, λ bloc design is a collecion of -subses of [n such ha every r elemens are conained in exacly λ subses In [13, Rands proved he following generalizaion of Erdős-Ko-Rado heorem: given a r n,, λ bloc design H and 0 < s < r, hen here exiss a funcion f, r, s such ha if H has an s-inersecing subhypergraph H, hen if n > f, r, s, he number of edges in H is a mos b s, which is he number of blocs hrough s verices Noe ha Erdős-Ko-Rado heorem corresponds o he very special case when H = [n and s = 1 Moreover, when s, r = 1, 2, his is an analogue of our Theorem 14, and when he bloc design is complee, i is similar o Theorem 15 Using ools developed in he previous secion, we can prove he following generalizaion of Manicam-Milós-Singhi conjecure o designs Given an r n,, λ design H, for j = 1,,, le d j be he number of blocs conaining a fixed se of j elemens Obviously r j r j λ j d r = λ, and by double couning, d j = n j Theorem 51 Le, r, be posiive inegers wih r 2, n > C 3+3 for sufficienly large C and le { } X [n be a weigh assignmen wih X [n 0 Then for a given r n,, λ design H, he number of blocs B wih X B,X [n 0 is a leas r r λ d = he elecronic journal of combinaorics , #P37 16

17 I would be ineresing if one can remove he condiion r 2 in his saemen This will give a resul generalizing our Theorems 14 and 15 for a weaer range of n The only addiional ingredien needed o prove he above heorem is he following fac For wo disjoin verex subses A = a and B = b of a r n,, λ design, he number of edges conaining every verex from A while no conaining any verex in B is eual o n r b r n a b r a n r r r a a λ We will omi any furher deails here and will reurn o his problem in he fuure In Secion 3, we gave an example of infiniely many r-uniform n-verex hypergraphs wih eual codegrees and n r 3 no having he MMS propery, based on he assumpion ha here are infiniely many Mersenne primes Since he larges nown Mersenne number has more han en million digis, our example already gives uncondiionally a huge hypergraph wih n cubic in r Sill i would be ineresing o consruc infiniely many such hypergraphs direcly, wihou relying on he exisence of Mersenne primes In Secion 4, we proved wo addiional generalizaions of he Manicam-Milós-Singhi conjecure Boh resuls can be regarded as he analogues of he Erdős-Ko-Rado heorem on he -inersecing families for sufficienly large n I would be ineresing o deermining he exac range for which hese heorems hold For example when = 1, Theorem 15 only gives n > 6 while we now from [12 ha i is rue already for n linear in Acnowledgmen We would lie o han Ameerah Chowdhury for bringing o our aenion a Maniam-Milos-Sighi conjecure for vecor spaces and for sharing wih us her preprin on his opic We also would lie o han he anonymous referees for carefully reading our manuscrip and providing many consrucive and helpful commens Noe added Afer his paper was submied, we learned ha Ihringer [7 recenly proved ha for large, Conjecure 13 holds for n 2 This bound is sharp and seles our uesion a he end of Secion 2 for large References [1 N Alon, H Huang, and B Sudaov, Nonnegaive -sums, fracional covers, and probabiliy of small deviaions, J Combin Theory Ser B, , [2 A Chowdhury, A noe on he Manicam-Milós-Singhi conjecure, European J Combin, 35C 2014, [3 A Chowdhury, G Saris, and S Shahriari, The Manicam-Milos-Singhi Conjecures for Ses and Vecor Spaces, preprin [4 P Erdős, C Ko, and R Rado, Inersecion heorem for sysem of finie ses Quar J Mah Oxford Ser, , [5 P Franl, On he number of nonnegaive sums J Combin Theory Ser B, o appear he elecronic journal of combinaorics , #P37 17

18 [6 S Hare, D Solee, A branch-and-cu sraegy for he Manicam-Milós-Singhi Conjecure, preprin arxiv: [7 F Ihringer, A noe on he Manicam-Milós-Singhi Conjecure for vecor spaces, preprin arxiv: [8 N Manicam, On he disribuion invarians of associaion schemes PhD hesis, Ohio Sae Universiy, 1986 [9 N Manicam and D Milós, On he number of non-negaive parial sums of a nonnegaive sum Collo Mah Soc János Bolyai, , [10 N Manicam and N Singhi, Firs disribuion invarians and EKR heorems J Combin Theory Ser A, , [11 G Marino and G Chiaseloi, A mehod o coun he posiive 3-subses in a se of real numbers wih non-negaive sum European J Combin, , [12 A Porovsiy, A linear bound on he Manicam-Milós-Singhi Conjecure, preprin arxiv: [13 B Rands, An exension of he Erdős-Ko-Rado heorem o -designs J Combin Theory Ser A, , [14 M Tyomyn, An improved bound for he Manicam-Milós-Singhi conjecure European J Combin, , A Missing proofs from Secion 42 Throughou his secion we use ha for a > b, a bb [ a b a bb+b Proof of Lemma 46: We may rewrie he lef hand side of he ineualiy as [ n 2 [ n [ n = [ n 1 1 dimx L=0 dimx L 1 2 b j j=0 dimx L=1 dimx L=j dimx L= 1 Here we le b j = [ [ n 2+j Noe ha 2+j dimx L 1 = X L 1 Since n > C, his implies 2 b j j=0 X L =j 1 24 [ 2 [ n [ n [ 2 [ n 3 For a fixed ineger 0 y 1, denoe by D y he number of nonnegaive - dimensional subspaces Z wih dimz L = y If D y > [ hen we are done Oherwise he elecronic journal of combinaorics , #P37 18

19 assume D y [ for every y We esimae he following sum: dimz L=y,dim Z= X Z Since every nonnegaive -dimensional subspace conribues o he sum a mos [, i is a mos [ D y [ [ By double couning, he above sum also euals y j=0 β y,j dimx L=j Here for a -dimensional subspace X wih dimx L = j, β y,j denoes he number of -dimensional subspaces Z such ha X Z and dimz L = y There are j y 1 ways o exend X L o Z L Le Q = span{x, Z L}, and y j y y 1 R = span{x, L} Then dim Q = +y j, dim R = 2 j, and Q R The nex sep is o exend Q o Z such ha Z R = Q The number of ways is eual o n 2 j n + y 1 +y j 1 Noe ha his is only nonzero for j + y, in his case β y,j is he produc of hese wo expressions, which is roughly yy j+n +j y Since + j y 0, i is increasing in j for large n Also noe ha b j is decreasing in j Le γ = γ 0,, γ 2 be he uniue soluion of he sysem of euaions b = γ β, hen by Lemma 41 2 b j j=0 dimx L=j 2 2 = β y,j γ y = j=0 y=j 2 γ y y=0 [ [ I is easy o chec ha β y,y n and so [ [ n 1 n 2 + y b y /β y,y = y y j=0 [ n [ n dimx L=j β y,j 2 γ y y=0 2 y=0 / n dimx L=j b y β y,y [ n 1 / n 1 Therefore 2 b j j=0 X L =j [ [ n [ 1 1 [ n n [ 1 1 n, ha for n > C conradics 3 he elecronic journal of combinaorics , #P37 19

20 Proof of Lemma 47: We will [ show ha every -subspace whose weigh is larger han 1 w P is conained in a leas nonnegaive edges, oherwise here are already more 2[ han [ P iself Suppose here are S negaive edges conaining w P, which are denoed by e 1,, e S as -dimensional subspaces And e S+1,, e [ are he oher hus nonnnegaive edges conaining w P We have [ i=1 P X e i = S i=1 P X e i + w P S + w P [ i=s+1 P X e i [n [ S [ [ n Here we used ha here are a mos 3 [ verices X whose weigh is larger han w P bu always 1, and he number of imes every such weigh appear in he sum is a mos [ 1 1 If S [ 1 2[ 1, hen he above expression is a mos [ n [ w P 1 2 [ which can be furher bounded by [ n [ 2 [ w P 3 [ [ w P 3 [ 1 n < 1 [ n 1, w P 1 12 [ 2 The firs ineualiy is because > 2 and 2, so [ 7, and also because n > C for large C Since we also have X P = w P 1 Lemma 46 for L = P immediaely gives > [ nonnegaive edges Therefore we can assume ha for he 3 [ verices wih larges weighs, he number [ 1 of nonnegaive edges conaining each such verex is a leas Using he union 2[ bound, he number of nonnegaive edges is a leas [ [ [ n which is also larger han [ [ 3 2 [ n when n > C 2 n 1 3[ [ n, he elecronic journal of combinaorics , #P37 20

21 Proof of Lemma 48: We consider all he [ -dimensional subspaces conaining [, similarly as before suppose here are S 2[ 1 negaive edges e 1,, e S and nonnegaive edges e S+1,, e [, we ge = [ Z,dim Z= X Z,X [ S i=1 X e i,x [ S [ [ n [ n 2 + [ i=s+1 X e i,x [ [n [ S S [ 1 [ 4 [ [ 1 [ 4 [ 3 [ n [ n 1 1 The firs ineualiy is by bounding he 3 [ larges weighs in he second sum by 1 and 1 he res by I also uses he fac ha wo -dimensional subspaces are conained in 4[ 2 a mos [ 1 -dimensional subspaces For n > C, we have 1 [ n 1 5 [ [ Z,dim Z= X Z,X [ We also have X [ = w [ = 1 Now we once again can apply Lemma 46 for L = [ o show he exisence of > [ nonnegaive edges Proof of Lemma 49: Noe ha if every -dimensional subspaces conaining [ is nonnegaive, his already gives [ nonnegaive edges and he lemma is proved So we may assume ha here is a negaive edge f as -dimensional subspace conaining [ wih X f < 0 Suppose he larges weigh ouside he edge f is w Q, where dimq f 1 Now we define new weighs w, such ha { [ = /w Q if X f oherwise Then for every X f, 1 and w Q = 1 Now we consider all he [ -dimensional subspaces conaining Q As usual, assume ha S of hem has negaive sum according o he elecronic journal of combinaorics , #P37 21

22 [ w If S 1 3[ 2, we have he following esimae: Noe ha since X f < 0, we have Q Y,dim Y = X Y,X Q w X [n [ S + S [ n [ [ n 12 X Q w X = X Q /w Q X f /w Q + [ 1 [ 2 If we apply Lemma 46 for L = Q and he new weighing = w X /2[ 2, we ge > [ nonnegaive edges for weigh w Noe ha every such nonnegaive edge canno share wih f a common -dimensional subspace, oherwise is oal weigh is a mos [ 1 [ < 0 Hence hese nonnegaive edges are also nonnegaive edges for he original weighing w By he above discussion, i remains o consider he case S < 1 2 [ 3 [ n Then here are a leas 2 3 [ [ n nonnegaive edges conaining Q, ogeher wih he [ [ n nonnegaive edges conaining [ Since [ and w Q have codegree a mos [ n 1 1 [ n 1 6 [, we have in oal more han [ nonnegaive edges he elecronic journal of combinaorics , #P37 22