Sperner s Theorem and a Problem of Erdős, Katona and Kleitman

Transcription

1 Sperer s Theorem d Problem of Erdős, Kto d Kleitm Shgi Ds Weyig G Bey Sudov Abstrct A cetrl result i extreml set theory is the celebrted theorem of Sperer from 98, which gives the size of the lrgest fmily of subsets of [] ot cotiig -chi F F. Erdős exteded this theorem to determie the lrgest fmily without -chi F F... F. Erdős d Kto, followed by Kleitm, sed how my chis must pper i fmilies with sizes lrger th the correspodig extreml bouds. I 966, Kleitm resolved this questio for -chis, showig tht the umber of such chis is miimized by tig sets s close to the middle level s possible. Moreover, he cojectured the extreml fmilies were the sme for -chis, for ll. I this pper, mig the first progress o this problem, we verify Kleitm s cojecture for the fmilies whose size is t most the size of the middle levels. We lso chrcterize ll extreml cofigurtios. 00 Mthemtics subject clssifictio: Primry 05D05 Itroductio Sperer s Theorem is cetrl result i extreml set theory, givig the size of the lrgest fmily of sets ot cotiig -chi F F. Erdős lter exteded this theorem to determie the lrgest fmily without -chi F F... F. A turl questio is to s how my -chis must pper i fmily lrger th this extreml boud. More precisely, we cosider the followig problem, first posed by Erdős d Kto d the exteded by Kleitm some fifty yers go. Give fmily F of s subsets of [], how my -chis must F coti? We deote this miimum by c, s, d determie it for wide rge of vlues of s. This provides qutittive stregtheig of the Erdős result o the size of -chi-free fmilies. We shll ow discuss the bcgroud of Sperer s Theorem d this problem further, before presetig our ew results.. Bcgroud Extreml set theory is oe of the most rpidly developig res i combitorics, hvig pplictios to other brches of mthemtics d computer sciece icludig discrete geometry, fuctiol lysis, umber theory d complexity. The typicl extreml problem hs the followig form: how lrge c structure be without cotiig some forbidde cofigurtio? A clssicl exmple, cosidered by my to be the strtig poit of extreml set theory, is theorem of Sperer [3]. A tichi Deprtmet of Mthemtics, UCLA, Los Ageles, CA, Emil: shgi@ucl.edu. Deprtmet of Mthemtics, UCLA, Los Ageles, CA, Emil: wg@mth.ucl.edu. Deprtmet of Mthemtics, ETH, 809 Zürich, Switzerld d Deprtmet of Mthemtics, UCLA, Los Ageles, CA Emil: bejmi.sudov@mth.ethz.ch. Reserch supported i prt by SNSF grt d by USA-Isrel BSF grt.

2 is fmily of subsets of [] tht does ot coti sets F F. Sperer s Theorem sttes tht the lrgest tichi hs / sets, boud tht is esily see to be tight by cosiderig the fmily of sets of size. This celebrted result ejoys umerous pplictios d hs my extesios, my of which re discussed i Egel s boo [4]. Oe prticulr extesio, due to Erdős [5], shows tht the size of the lrgest set fmily without -chi, tht is, -sets F F... F, is the sum of the lrgest biomil coefficiets, M =. Whe =, we recover Sperer s i= i Theorem. Our problem is wht we refer to s Erdős Rdemcher-type extesio of Erdős theorem, me we ow expli. Argubly the most well-ow result i extreml combitorics is theorem of Mtel [] from 907, which sttes tht -vertex trigle-free grph c hve t most 4 edges. I upublished result, Rdemcher stregtheed this theorem by showig tht y grph with 4 edges must coti t lest trigles. Erdős [6] the exteded this to grphs with lier umber of extr edges, d i [7] studied the problem for lrger cliques. More geerlly, for y extreml problem, the correspodig Erdős Rdemcher problem ss how my copies of the forbidde cofigurtio must pper i structure lrger th the extreml boud. I the cotext of Sperer s Theorem, this problem ws first cosidered by Erdős d Kto, t sets must coti t lest t -chis. Kleitm who cojectured tht fmily with / [0] cofirmed the cojecture, d, i fr-rechig geerliztio, showed the miimum umber of -chis i fmily of y fixed size is obtied by choosig sets of size s close to s possible. He the cojectured see [8, 0] tht the sme fmilies miimize the umber of -chis, problem tht hs remied ope for erly fifty yers. Cojecture.. The umber of -chis i fmily is miimized by choosig sets of sizes s close to s possible.. Our results I this pper we study these Erdős Rdemcher-type extesios of the theorems of Sperer d Erdős. We beg by cosiderig the cse of -chis, d determied the miimum umber of - chis i fmily of y umber of sets. Lter, we discovered Kleitm hd erlier obtied the sme result. However, through slightly more creful clcultios, d by itroducig dditiol rgumet, we re ble to chrcterize ll extreml fmilies, s give below.. Let r Theorem.. Let F be fmily of subsets of [], with F = s / N be the uique hlf-iteger such tht r i= r i < s r i= i. The F miimizes the umber of -chis r if d oly if the followig coditios re stisfied:. For every F F, r r.. For y A [] with r A r, we hve A F. 3. If s r i= r i, the {F F : = ± r} forms tichi. 4. If s r i= r i, the {F / F : = ± r} forms tichi.

3 Our mi results verify Cojecture. for fmilies of certi sizes. To begi with, recll tht Erdős showed the lrgest fmily without -chis cosists of the middle levels of the hypercube, whose size we deote by M. If we were to dd oe set to this fmily, the best we could do would be to dd it to the th level, i which cse we would crete /! -chis. Ideed, we show tht every dditiol set must cotribute t lest this my ew -chis, d the bove costructio shows this is tight whe our extreml fmily is cotied withi the middle levels. Theorem.3. If F is set fmily over [] of size s = M t, the F cotis t lest t /! -chis. We re the ble to exted our rgumet to wor for lrger set fmilies, obtiig result tht is tight whe the extreml fmily is cotied withi the middle levels. Theorem.4. Provided 5 d 6, if F is set fmily over [] of size s = M t, the the umber of -chis i F is t lest / / /! t!. / I both cses, we ctully obti stroger results see Theorems 3. d 4. respectively, providig stbility versios of the bove theorems, showig tht if fmily hs close to the miimum umber of -chis, it must be close i structure to the extreml exmple. These stbility results re of iterest eve i the cse =, s oe does ot obti y stbility from the Kleitm proof for -chis. We the use the stbility results to show tht whe the bove bouds re tight, the extreml fmilies re exctly s i Theorem...3 Outlie d ottio The remider of this pper is orgized s follows. Sectio cotis proof of Theorem.. I Sectio 3, we prove Theorem.3, d the i Sectio 4 prove Theorem.4. I the fil sectio we preset some cocludig remrs d ope problems. Appedix A cotis the proof of techicl propositio eeded for Theorem.. We let [] deote the set of the first itegers. For groud set X d iteger i, we deote the fmily of i-subsets of X by X i = {Y X : Y = i}. We let M = / i= / i be the size of the middle, d thus lrgest, levels. Give fmily F of subsets of [], we let F i = F [] i deote those sets i F of size i. The l-shdow of fmily is give by l F = {G : F F, G F, G = l}. For subset F [], we defie mf = mx{, }. Give set fmily F, c F deotes the umber of -chis i F. For y N d 0 s, we let c, s deote the miimum of c F over ll fmilies F of s subsets of []. Whe =, if we hve two fmilies F d G, the we let c F, G deote the umber of -chis with oe set from F d oe set from G. Coutig -chis I this sectio we will prove Theorem., chrcterizig those fmilies tht miimize the umber of -chis. We essetilly show tht it is optiml to te sets of sizes s close to s possible. The theorem the prescribes how the boudry sets c be distributed. 3

4 Sperer s Theorem shows tht the lrgest tichi is give by oe of the middle levels, tht is either ll sets of size or ll sets of size. Obviously, tichi miimizes the umber of -chis, s it hs oe. This theorem is the turl extesio of Sperer s Theorem, s it shows tht to costruct fmily of y size tht miimizes the umber of -chis, oe should strt by tig sets of size, the sets of size, the, d so o util oe hs fmily of the desired size. As we shll show, these fmilies re optiml, d so we my deote the umber of -chis i the first s such sets by c, s. The ide behid the proof is s follows. If our fmily F cotis set F tht is too fr wy from the middle i.e. > r, the we will show tht we c shift F closer to the middle d decrese the umber of -chis. Oce we hve our fmily cotied i the r middle lyers, simple coutig rgumet will give the chrcteriztio of extreml fmilies. As the shiftig process is essetilly the sme s i Kleitm s proof i [0], we relegte the proof of the followig propositio to Appedix A. Propositio.. Let F be fmily of s > / subsets of [] miimizig the umber of -chis. If A F is of mximl crdility, with A = m, the for y B A, B m, we hve B F. Assumig this propositio, we shll proceed to prove Theorem.. Proof of Theorem.. We prove the theorem by iductio o s. For the bse cse, we te s = /. By Sperer s Theorem, it follows tht y fmily F of this size tht miimizes the umber of -chis must be tichi. It is well ow tht the oly tichis of this size re the fmily of ll sets of size, or the fmily of sets of size. It is esy to see tht these fmilies re the oly oes stisfyig Properties through 4, with r = 0 or depedig o whether is eve or odd respectively. For the iductio step, ssume s > /, d let F be optiml fmily of size s. Suppose Property were ot stisfied. Sice F d F = {[] \ F : F F} hve the sme umber of -chis, we my ssume there is lrgest set F F with = t for some t > r. By Propositio., it follows tht for every G F, G t, we hve G F. Hece F is i t lest t t i= i -chis i F. Sice F \ {F } is fmily of s sets, there re t lest c, s -chis i F ot ivolvig F. Thus the umber of -chis i F is t lest c, s t t i= i > c, s r r i= i c, s, d so F cot be optiml, givig cotrdictio. Hece if F is optiml, ech F F hs r r, d so Property is estblished. Now cosider the cse r i= r i < s r i= r i. Sice s > r i= i, d i light of r Property, it follows tht there exists some F F with = ± r; by symmetry, we my ssume = r. By Propositio., F must be cotied i r r i= i = c, s c, s -chis with sets i F of sizes betwee r d r. If F is cotied i y -chis with sets of size r, the by iductio it follows tht F hs more th c, s -chis, cotrdictig the optimlity of F. Thus F is icomprble to the other sets i F of sizes ± r. Removig F, we fid tht F \ {F } must lso be optiml, d thus Properties d 3 follow. Filly, suppose r i= r i s r i= r i. By Property, we ow ll sets i F hve sizes 4

5 betwee r d r. Let H = r [] i= r i be the fmily of ll subsets of [] of sizes betwee r d r, d let G = H \ F be those sets ot i F. We hve c F = c H c G, H c G. c H depeds oly o r, d hece o s, d is idepedet of the structure of F. We hve c G, H = c {G}, H = G G r G. i i G G G G G i= r i= G The pretheticl term is mximized whe G = ± r, d so c G, H is mximized whe for every G G we hve G = ± r. Filly, c G is miimized whe G is tichi, i which cse c G = 0. Both of these coditios re stisfied by the costructio outlied t the begiig of this sectio, d hece must lso be true of y other extreml fmily. Thus to miimize c F, F must coti ll sets of sizes betwee r d r, d G = [] [] r \ F should be r tichi, estblishig Properties d 4. This completes the iductio step, d with it the proof of Theorem.. 3 Coutig -chis We ow see similr result for -chis, d thus to me some progress o Cojecture.. I this sectio we verify the cojecture whe the umber of sets is t most tht i the middle levels, d i the ext sectio we shll exted the result to the middle levels. Note tht if we te ll M sets i the middle levels, tht is of sizes betwee d, d the dd t sets of size Hece Theorem.3 is tight whe 0 t, we would crete precisely t /! -chis. /. We shll i fct prove the followig stroger theorem, which provides stbility result. I the followig ottio, we let r {, } be such tht the sets i the middle levels hve sizes betwee r d r, d for set F, we defie mf = mx{, }. Theorem 3.. Let F be fmily of subsets of []. The the umber of -chis i F is bouded by /r c F F i / r mf!!. i / r i=/ r F F: / r Proof. We prove the theorem by iductio o F. If F = 0, the there is othig to show, s the desired lower boud is egtive. For the iductio step, we begi by otig tht for every set F F with r, F c be i t most mf! -chis. If ot, the we could remove F, d pplyig the iductive hypothesis to F \ {F }, we would hve the desired iequlity. We ow use LYM-type iequlity, coutig the umber of -chis i our fmily by cosiderig permuttios. We sy tht permuttio σ S cotis set F [], deoted F σ, if {σ, σ,..., σ} = F ; tht is, F is iitil segmet of σ. Note tht if σ cotis sets F, F,..., F, the those sets must form -chi. For y set F [], we let S [F ] = {σ S : 5

6 F σ}. Sice every permuttio cotiig m sets cotributes m m to the right-hd side of the sum below, it follows tht! S [F ] i=s [F i ]. F F F F... F F As the secod sum is over ll -chis i our fmily F, this iequlity will llow us to boud the umber of -chis. Note tht for y F F, we hve S [F ] =! [] \ F!, d for -chi F F... F, i= S [F i ] =! i= F i \ F i! [] \ F! gives the umber of permuttios cotiig the -chi. We shll ssocite every -chi i F with either its miimum or mximum set, depedig o which is further wy from the middle level. For F F with <, let CF = {F F... F : F < }, d if F F with, let CF = {F... F F : F }, d, for coveiece, defie CF = CF. Note tht we hve prtitioed the set of -chis {F F... F : F i F} ito the disjoit sets {CF } F F. We c thus rewrite iequlity s follows:! F F S [F ] F F... F CF i=s [F i ]. To boud i= S [F i ] ppropritely, we require tht d [] ot be members of our fmily. This is give by the followig lemm. Lemm 3.. If F M, d F miimizes the umber of -chis, the / F d [] / F. Proof. Suppose we hd [] F. Sice, there must be some F / F. We decrese the umber of -chis i F by replcig [] with F, sice y ew -chi ivolvig F ws -chi with [] before. Similrly, if F, we c replce it with y set [] F / F. Note tht M =. Thus, if F > M, the either we hve ll subsets of [], or our fmily is missig just oe set, i which cse s we explied bove it is best to remove either or []. I either cse, the boud i Theorem 3. remis true. We ow ssume for ll F F. If we fix F, F < /, the we mximize i= S [F i ] = F! i= F i \ F i! [] \ F! by tig F i \ F i = for i, d so i= S [F i ]! F!. The sme holds true if we isted fix F, F /, d F F!! thus i= S [F i ] F! F!. We c uify both bouds i the form. mf! Moreover, by defiitio we must hve CF = for y F with r. Hece we split our sum bsed o how compres to r. Dividig through by!, iequlity leds to CF = Σ Σ Σ 3,! where Σ = F F / r mf = 6 /r i=/ r F i i,

7 d Σ = /r F F / =r /r /r Σ 3 = CF F / r F /r =! F F / r /r CF. mf /r F F / =r /r CF,! Note tht / =r CF = c F / r CF, d so, substitutig i Σ, we obti c F! /r /r i=/ r F i i F F: / r CF! Now, sice = s s lrge s possible, which, by our iductive hypothesis, is mf iequlity bove gives /r, it follows tht whe mf mf mf /r /r /r. /r r, r mf. Hece the summd is miimized whe CF is c F! /r /r i=/ r F i i Rerrgig gives the desired boud /r c F F i / r i / r i=/ r!. Substitutig this ito the F F: / r! /r F F: / r mf /r. mf!., Give Theorem 3., it is esy to deduce Theorem.3. I fct, we re ble to chrcterize ll extreml fmilies. Proof of Theorem.3. Suppose we hve fmily of sets F, with F = M t. Note tht the cotributio ech set F F mes to the right-hd side bove is /r /r! if r r, d mf! otherwise. This cotributio icreses with, d so to miimize the right-hd size we eed ll sets to stisfy r. Moreover, sice the biomil coefficiets i re miimized over / r i / r whe i = / ± r, y extreml fmily must coti ll sets of sizes betwee r d r, with the remiig sets hvig size ± r. It is esy to see tht such collectio of sets gives c F t /r! bove, s required. To clssify the extreml fmilies, ote tht we lredy ow we must hve ll sets of sizes betwee r d r. If r =, this cosists of the middle levels, givig M sets. Hece we 7

8 hve t sets of size ± r. To obti equlity i, we must hve every chi pssig through either or sets of F. As ll the sets i the middle levels re i F, the chi c coti t most oe set from F of size ± r. From this, we deduce tht {F F : = ± r} must be tichi. If r =, we ow the middle levels re full, d we hve /r t sets of size ± r. I order to obti equlity i, we must therefore hve every chi pss through t lest oe set i F of size ± r. Hece {G / F : G = ± r} must form tichi. I prticulr, we ote tht the extreml fmilies re exctly the sme s for Theorem.. We remr tht Theorem 3. is stbility result for Theorem.3, s our boud o c F icreses if we re missig sets with r, or hve sets with r. Moreover, the stbility estimtes we obti c lso be tight. For istce, if we replce l sets of size ±r with sets of size r, Theorem 3., together with some simple computtios, shows tht we should gi t lest l /r! extr -chis. Moreover, it is esy to chec tht we gi precisely tht my -chis i the cse whe our fmily icludes ll of the r -sets d oe of the r -sets i the shdow of the r -sets which were dded. Similrly, if we replce l sets of size r with sets of size r, the theorem shows tht we must gi t lest r /r l /r! extr -chis. This is tight gi, if our fmily icludes ll the sets of size r cotiig y of the replced sets. As we remred erlier, these stbility results re ew eve i the cse =. 4 Lrger fmilies While Theorem.3 provides tight boud o the umber of -chis pperig i fmilies cotied withi the middle levels, it uderestimtes the umber of -chis pperig i lrger fmilies. This is becuse i our clcultios we ssumed every -chi hd steps F i \ F i of size, s this mximizes the umber of permuttios cotiig the -chi. However, whe we re worig with the middle levels, we lso hve -chis with lrger step of size, d so we shll hve to me our rgumet more robust i order to hdle these chis. However, there is oe dditiol difficulty. Recll tht the umber of permuttios cotiig -chi F F... F is give by i= S [F i ] = F! i= F i \ F i! F!. If we fix F, sy, the we would hope tht for -chis ivolvig lrger step, the lrgest this c be is to hve oe step of size, d hve ll the other steps hve size, s this is precisely the type of -chi tht ppers i our extreml fmilies. Such -chis re i F! F! permuttios. Ufortutely, chi with steps of size d oe step of size F so tht F = is cotied i F! F! permuttios, which is lrger th the boud we require. However, recll tht we ssig -chis to either F or F, depedig o which is further from the middle. Thus, uless F =, we would ssig the bove -chi to F, d so we would be fixig F d ot F. Hece the oe cse we eed to void is hvig -chi strtig with set of size d edig with set of size. We shll lter provide seprte rgumet to show tht there cot be y sets of size i extreml fmily, thus bypssig this problem. I the mewhile, for the purposes of our stbility results, we shll ssume there re o sets of size. We ow require two rgumets - oe to boud the umber of -chis with lrger steps, d oe to boud the totl umber of -chis. 8

9 We itroduce the followig ottio for the remider of this sectio. Give d, we let =, so tht the middle levels re those sets of sizes betwee d, d the st middle level hs sets of size. For set fmily F, we let CF be the set of -chis i F. We prtitio these ito two subsets: C F re those -chis F F... F with F i \ F i = for ll i, d C F = CF \ C F those -chis with lrger step. We let CF, C F d C F deote the umber of -chis i these subsets respectively. As i Theorem.3, we will gi idetify -chi with oe of its edpoits F or F, depedig which is further from the middle level, givig the prtitio {CF } F F of CF. These sets will gi be prtitioed ito C F d C F, depedig o whether or ot the -chis hve step of size t lest. Filly, CF, C F d C F represet the sizes of the correspodig sets of -chis. 4. Coutig -chis with lrger steps We begi by showig tht lrge fmilies must coti umber of -chis with step of size t lest. The followig propositio lso provides some stbility, which we shll require to show tht extreml fmily cot coti y sets of size. Propositio 4.. Let F be set fmily of size F = M t, with for ll F F, d with t lest t sets missig from the middle levels. The C F t t!. Proof. We prove the sttemet by iductio o t t 0, otig tht we must hve t 0. The bse cse of t t = 0 is trivil, s i this cse the right-hd side is o-positive. The proof will ow ru log very similr lies to tht of Theorem.3, d we shll just me few chges to cout oly those chis with lrge step. To begi with, whe we re coutig sets d -chis i permuttios, we oly wt to cosider those -chis with lrge step. To esure this, we shll ot cout -chis tht pper cosecutively i some permuttio. Tht is, if σ S cotis the sets F F... F s for some s, we will cout F F... F F, but ot F F... F. Thus every -chi we cosider is boud to hve some step of size t lest. If s, the the umber of such chis is s s, d sice s s s, it follows tht! F F F F S [F ] S [F ] F... F C F F... F C F i=s [F i ] i=s [F i ]. As before, we ow see to mximize the terms i= S [F i ]. Provided we hve for ll sets F F, if we fix oe of the edpoits of the chi, the umber of permuttios it is cotied i is mximized whe we hve oe step of size, d ll the other steps of size. Thus we c boud i= S [F i ] by! F! or F! F!. Dividig through by!, we hve tht F F 9 C F mf!.

10 Now, by defiitio, we must hve C F = 0 for ll sets F i the middle levels; tht is, with. Let ˆF = {F F : or } be those sets outside the middle levels. Thus i= F i i F ˆF C F mf We my ssume tht for every set F ˆF, C F! =, sice otherwise we my remove F from F d re the doe by iductio. Hece, sice mf for ll F F, the pretheticl term i the secod sum is lwys o-egtive, d so the right-hd side is miimized by replcig by, givig d so = C F! i= i= i= F i i F ˆF F i i F ˆF F i i i= ˆF! C F mf! C F C F!! F i ˆF. i If the middle levels were full, the the first sum would equl. Sice we must hve t lest t sets missig from the middle levels, the right-hd side is miimized whe there re exctly t sets missig, ll of size. I this cse, ˆF = t t, givig C F! t t = t,.! t. As =, we hve, d so multiplyig through by! desired boud. 4. Coutig ll -chis gives the As Propositio 4. offers us some cotrol over the umber of chis with lrge steps, we c ow proceed to boud the totl umber of -chis i F. Agi, our result provides somes stbility, s we shll require to forbid sets of size. Theorem 4.. Let F be set fmily of size F = M t, with for ll F F, d with t lest t sets missig from the middle levels. The CF! t t!. 0

11 Proof. We prove the theorem by iductio o t t 0, d gi must hve t 0. The bse cse of t t = 0 follows from Theorem.3, s whe F = M = M, the right-hd side bove is less th the lower boud for t = i Theorem.3. We my ow ssume tht y set F ot i the middle levels is cotied i t most! -chis. If ot, the we my remove F from F, thus decresig t by. Applyig the iductive hypothesis to F \ {F } d ddig the -chis ivolvig F the gives the requisite umber of -chis. We oce gi see to boud the umber of -chis i our fmily by coutig sets d -chis i permuttios, except this time we shll cosider ll -chis pperig i the permuttios. A permuttio with s sets gives rise to s -chis, d sice for ll s we hve s s, it follows tht! S [F ] F F = F F S [F ] F... F CF F... F CF i=s [F i ] 3 i=s [F i ]. To mximize i= S [F i ], sice we hve o sets of size, we should gi te ll the gps to be s smll s possible. Those chis i C F ll hve steps of size, while those i C F should hve oe step of size, d the rest of size. Dividig by! gives F F mf C F C F.! mf By defiitio, if, we must hve C F = C F = 0, d if =, the C F = 0. Thus we hve three types of -chis to cosider: those i C F for =, those i C F for F ˆF = {F F : or }, d those i C F for F ˆF. Splittig our sums thus, we obti Sice i= F ˆF F i i = C F! F ˆF C F. mf! = C F = CF F ˆF C F F ˆF! C F, mf we c substitute this expressio ito the secod sum, d redistribute, to obti CF! i= F i i Σ Σ, C F!

12 where Σ = F ˆF Σ = F ˆF C F! mf C F mf! C F, d! C F.! The followig lemms, whose proofs we defer to the ed of this subsectio, llow us to boud these sums. Lemm 4.3. For every F ˆF, we hve C F! mf C F!. Lemm 4.4. For every F ˆF, we hve C F mf! C F. We ow replce our summds with these lower bouds, obtiig CF F i! i= i F ˆF C F F ˆF! = i= F i i ˆF C F!!. We c use Propositio 4. to lower boud C F. Moreover, s, it follows tht ech set i ˆF, whose size is ot, hs greter weight th y set i the middle levels. Thus, the right-hd side is miimized whe we fill the middle levels s much s possible. If we were to hve the full middle levels, the first sum would be equl to. However, s we must hve t lest t sets missig from the middle levels, it is best to hve exctly t sets of size missig, resultig i CF t t t! t t = A t A t,!!!

13 where, fter simplifyig the biomil expressios, we fid A = = A =, d = Multiplyig through by.! gives the desired boud. To complete the proof, we ow prove the two lemms. Proof of Lemm 4.3. Note tht mf = mf, d, by the sme toe, =. Sice =, d, s F ˆF, we hve mf, it follows tht mf. Thus the left-hd side of the iequlity is miimized whe we choose C F s lrge s possible. By defiitio, C F mf!. Mig this substitutio gives mf C F C F mf!!. Proof of Lemm 4.4. If mf =, the we hve equlity, so we my ssume mf. By iductio, we c ssume tht o set is i more th! -chis, givig boud o C F. Thus mf! C F!!, d so the fctor C F is o-egtive. As, we thus hve mf! C F mf! C F mf! C F.! 4.3 Forbiddig lrge sets Give the previous theorem, ll tht remis is to show tht extreml fmily cot coti sets of size. The ide behid this is s follows. A set of size hs very lrge shdow i the middle levels. I order for this set to ot coti too my -chis, we must therefore be 3

14 missig lot of sets i the middle levels. By Theorem 4., it the follows tht the remider of the fmily must coti my more -chis th it ought to. The relevt clcultios re give below. Propositio 4.5. Suppose 5 d 6, d let F be set fmily with = M t. If we do ot hve for ll F F, the c F >! t!. Proof. The sme proof s i Lemm 3. shows tht we my ssume we do ot hve or [] i F. Hece it suffices to show there re o sets of size i our fmily. Suppose towrds cotrdictio we hd some set F F with =. We my, by iductio, ssume tht F is i t most! -chis. Sice F cotis sets of size, there re! possible -chis tht F might be i which cosist of sets from the middle levels followed by F. Hece we must be missig lot of sets from the middle levels to prevet F from beig i too my -chis. The sets of size re cotied i the most such -chis, so if we re missig t sets, we must hve t!!. Solvig for t gives t. We ow remove from F ll sets of size, thus losig t most sets, d pply Theorem 4. with the bove vlue of t. I the theorem, the umber of -chis is govered by the expressio t t. We re decresig t by t most, but icresig t by t lest, resultig i et gi i the previous expressio of t lest. Sice 6, we hve 3. If is some costt, the the first term is t lest cubic i, while the term we re subtrctig is qudrtic, sice i this cse will lso be costt. Oe the other hd, if is lrge, the first term will be t lest lrge power of, while the term we subtrct is t most cubic i. Give 5, some simple but tedious clcultios show tht i either cse, hvig set of size icreses the umber of -chis our fmily must coti. Note tht the coditio 6 is er-optiml, sice if = 3, the M =, d so by volume cosidertios loe there must be extreml fmilies with sets of size t lest. Theorem.4 ow follows esily. Proof of Theorem.4. Sice 5 d 6, Propositio 4.5 shows tht for ll F F. We my the pply Theorem 4. with t = t d t = 0 to obti the boud CF! t!. 4

15 Recllig tht =, this is precisely the desired lower boud. We c gi deduce chrcteriztio of the extreml fmilies. Note tht we must hve t = 0 for the bove boud to hold, d so the middle levels must be full. I order to hve equlity i Lemm 4.4, we lso eeded mf = for ll F ˆF. If =, the the remiig t sets must hve size ±. To obti equlity i 3, every chi must pss through either or sets of F, d so we must hve {G / F : G = ± } formig tichi. If, o the other hd, =, the sets of size = crry greter weight th sets of size =. We c the redefie t bove to be the umber of sets missig i the middle levels d obti the sme result. Hece it follows tht we must hve ll sets i the middle levels, with the remiig sets i ˆF of size ±. I order to miti equlity i 3, every chi must pss through t most oe set F F with = ±, d so {F F : = ± } must be tichi. Thus, oce gi, the extreml fmilies re exctly the sme s those tht miimize the umber of -chis, s give by Theorem.. 5 Cocludig remrs d ope problems I this pper, we hve prtilly swered Kleitm s cojecture by showig tht the fmilies tht miimize the umber of -chis lso miimize the umber of -chis whe they occupy up to the middle levels. While we strogly believe the cojecture is true i geerl, we suspect ew ides re eeded to del with lrger fmilies. As the umber of levels grows with respect to, the umber of differet types of chis - i terms of the sizes of the steps betwee sets - grows rpidly, d these would ll eed to be cotrolled to obti precise result. I this directio, though, the sme methods we hve used bove c be pplied to show the followig: if we hve itegers α, α,..., α with i α i = l, the, provided F M l d the lrgest set i our fmily hs size t most mx i α i, the umber of -chis F F... F with F i \ F i α i is miimized by tig sets i the middle l levels. Cosiderig the cse of -chis, our pper hs focused o showig tht fmily with more th / sets must coti my -chis. A closely relted problem is to determie whether such fmily must hve y sets cotied i my -chis. This type of questio hs bee studied before i other settigs. For exmple, whe oe is cosiderig the umber of trigles i grph, Erdős showed i [6] tht y grph with 4 edges must coti edge i t lest 6 o trigles. It is well-ow d esy to see tht the hypercube, grph whose vertices re subsets of [], with two vertices djcet if they re comprble d differ i exctly oe elemet, hs idepedece umber. Chug, Füredi, Grhm d Seymour [] proved y iduced subgrph o vertices cotis vertex of degree t lest o log. It is ope problem to determie whether or ot this boud is tight the correspodig upper boud is O, d the swer to this questio hs rmifictios i theoreticl computer sciece. I the cotext of Sperer s theorem the bove problem hs egtive swer, which my be surprisig give the previous two exmples. For coveiece, let us ssume = m is odd, d cosider the followig set fmily. Let F = {F : / F, = m} {F : F, = m }. 5

16 This fmily cotis m m = / sets, d so we re ideed beyod the Sperer boud. However, it is esy to see tht the oly pirs of comprble sets re of the form {F, {} F } for every F F with = m. Hece ech set of the fmily is i oly oe pir of comprble sets. I fct, for this fmily we hve c F = c, F, so it is possible to hve extreml fmily with the comprble pirs distributed s evely s possible. Theorem. shows tht y fmily with m m sets must coti set i t lest two -chis. It is ope problem s to whether this is lso the lrgest fmily without set tht cotis two other sets d hece is the mximum set i two -chis. This cofigurtio is ow s -for, d the upper boud, which c be obtied usig similr rgumets s i Theorem.3, is /, s show by Kto d Trjá [9]. We fid most excitig the prospect of studyig Erdős Rdemcher-type problems i other settigs. Withi the cotext of Sperer s Theorem, pper of Qi, Egel d Xu [] studied extesio for multiset fmilies, where the sme set my be chose multiple times. I [], we derive Erdős Rdemcher-type stregtheig of the Erdős Ko Rdo Theorem. However, s oe c ivestigte similr extesios for y extreml result, there is truly o ed to the umber of directios i which this project c be cotiued. We hope tht further wor of this ture will led to my iterestig results d greter uderstdig of clssicl theorems i extreml combitorics. Note dded i proof: Durig the preprtio of this muscript, it cme to our ttetio tht Dove, Griggs, Kg d Serei [3] hve idepedetly obtied Theorem.3. Refereces [] F. R. K. Chug, Z. Füredi, R. L. Grhm d P. Seymour, O iduced subgrphs of the cube, J. Comb. Theory Ser. A , [] S. Ds, W. G d B. Sudov, The miimum umber of disjoit pirs i set systems d relted problems, rxiv.org: [3] A. P. Dove, J. R. Griggs, R. J. Kg d J. S. Serei, Supersturtio i the Boole lttice, to pper i Itegers. [4] K. Egel, Sperer Theory, Cmbridge Uiversity Press, 997. [5] P. Erdős, O lemm of Littlewood d Offord, Bulleti of the Americ Mthemticl Society 5 945, [6] P. Erdős, O theorem of Rdemcher-Turá, Illiois Jourl of Mth 6 96, 7. [7] P. Erdős, O the umber of complete subgrphs cotied i certi grphs, Mgy. Tud. Acd. Mt. Kut. It. Közl. 7 96, [8] P. Erdős d D. Kleitm, Extreml problems mog subsets of set, Discrete Mth , [9] G. O. H. Kto d T. G. Trjá, Extreml problems with excluded subgrphs i the -cube, Grph Theory, Spriger Berli Heidelberg 983,

17 [0] D. Kleitm, A cojecture of Erdős-Kto o commesurble pirs mog subsets of -set, Theory of Grphs, Proc. Colloq., Tihy 966, 5 8. [] W. Mtel, Problem 8, Wiudige Opgve 0 907, [] J. Qi, K. Egel d W. Xu, A geerliztio of Sperer s theorem d pplictio to grph oriettios, Discrete Applied Mthemtics , [3] E. Sperer, Ei Stz über Utermege eier edliche Mege, Mthemtische Zeitschrift 7 98, A The shiftig propositio I this ppedix, we prove Propositio., which ebles us to perform the shiftig ecessry for Theorem.. As metioed i Sectio, this is essetilly the sme shiftig rgumet used i the origil proof of Kleitm i [0]. We provide the proof here s the detils of the clcultios re ot icluded i Kleitm s pper. Proof of Propositio.. Suppose ot. Note tht we must hve m, otherwise there is othig to prove. Let l m be the miiml iteger such tht there exists lrgest set of size m with subset of size m l tht is ot i the fmily. Let A = {A F : A = m, l A F}, d let B = l A \ F. We c costruct uxiliry biprtite iclusio grph o A B, with edge A, B iff B A. Cosider first the cse where we hve mtchig M : A B, so tht for every set A A there exists set MA A, MA / F. We shift the fmily from F to F by replcig ech set A A by MA B, d clim tht this reduces the umber of -chis. Note tht if B = MA is ewly-itroduced set, d C F is set with C B, the we must hve hd C A s well. Thus the oly -chis tht we eed to cosider re those betwee the levels m d m l; we cll these itermedite chis. Suppose l >. By the miimlity of our choice of l, we must hve i A F for every i l. Thus the umber of itermedite chis i F tht we lose is t lest A l m i= i. O the other hd, ll sets of size m i F re the sets from F with i-shdow completely i F for ll i l. These sets cot be ivolved i y -chis with sets i B, d therefore we oly gi itermedite chis betwee the levels m d m l. The umber of such chis tht we gi is t most A l ml i= i. Sice l m, it follows tht ml i < m i for every i l, d hece the umber of -chis decreses. Thus we my ssume l =. If we hd A A d B F with B A, B = m, the upo shiftig to F, we lose the -chi B A d gi o pirs. Hece we my ssume A F =, so B = A. We ow clim tht the sets A A cot be ivolved i y -chis C A i F. Suppose to the cotrry we hd such -chi. Let x C be rbitrry elemet of C, d shift A to A \ {x} recll tht A \ {x} F. Shift the remiig sets i A by rbitrry mtchig from A = A \ {A} to B = A \ {A \ {x}}; we c do this by Hll s Theorem, sice every set i A hs t lest m eighbors i B, while ech set i B hs t most m eighbors. I this shifted set we hve lost the -chi C A, d hece F hs fewer -chis. 7

18 Hece we my ssume tht there re o -chis i F ivolvig sets i A. Thus i the shifted fmily F, sets i A will lso ot be i y -chis. Now, sice F > /, it follows from Sperer s Theorem tht there is some -chi C D i F. I F, we my lso shift D to some set i A, sice m implies A > A. As o set i A is ivolved i y -chi, this reduces the umber of -chis, which cotrdicts the miimlity of F. Therefore we coclude tht there cot be mtchig from A to B i the uxiliry biprtite iclusio grph, d so we will ot shift ll sets i A. Isted, we use the followig lemm, to be prove shortly, to fid collectio of sets to shift. Lemm A.. Let G be biprtite grph o U V with miimum degree δ U i U d mximum degree V i V. Suppose there is o mtchig from U to V. The there exist oempty subsets U U d V V with perfect mtchig M : U V d eu, V eu \ U, V U V. Our uxiliry grph stisfies the coditios of the lemm, with U = A, V = B, V = ml l < m l, d so we c fid collectio of sets A A d mtchig M : A B B s give by the lemm. Cosider the shifted fmily F where we replce the sets i A by the correspodig sets i B. As before, sice for every A A we hve MA A, we eed oly cosider the itermedite chis. Agi, by the miimlity of l, we ow tht A hs full shdow i F up util the lth shdow, d so the sme clcultio s before implies tht we remove more chis th we gi, d thus hve fewer itermedite chis i F. Hece it suffices to cosider oly the ew chis formed betwee levels m d m l. The umber of ew chis betwee these levels we gi is exctly ea \ A, B. O the other hd, we lose ll chis betwee A d l A F = l A \B. Thus the umber of chis we re losig is A m l ea, B. By the lemm, we hve A m l ea, B > A ml l ea, B ea \ A, B, d hece F hs fewer -chis th F, cotrdictig the optimlity of F. Thus if F miimizes the umber of -chis, d A is the lrgest set i F with A = m, the wheever B A with B m, we must hve B F s well. It remis to furish proof of Lemm A., which we ow provide. Proof of Lemm A.. As there is o mtchig from U to V, by Hll s Theorem there exists miiml subset U 0 U with NU 0 < U 0. Sice δ U, we must hve U 0. Let u U 0 be rbitrry elemet, d te U = U 0 \ {u}. By the miimlity of U 0, it follows tht NU U, d so we must hve NU = U. Set V = NU. Agi by the miimlity of U 0, for y subset X U, NX X, d so by Hll s Theorem there exists perfect mtchig M : U V. Now eu, V eu \ U, V = eu, V eu, V eu, V = eu, V V V = U V, where the secod equlity follows from the fct tht NU = V, d thus we hve the desired iequlity. 8