Short Proofs of the Kneser-Lovász Coloring Principle

Transcription

1 Short Proofs of the Keser-Lovász Colorig Priciple James Aiseberg 1,, Maria Luisa Boet 2,, Sam Buss 1,, Adria Crãciu 3,, ad Gabriel Istrate 3, 1 Departmet of Mathematics, Uiversity of Califoria, Sa Diego, La Jolla, CA , USA jaiseberg@math.ucsd.edu, sbuss@ucsd.edu 2 Computer Sciece Departmet, Uiversidad Politécica de Cataluña, Barceloa, Spai boet@cs.upc.edu 3 West Uiversity of Timişoara, ad the e-austria Research Istitute, V. Pârva 4, cam. 045B, Timişoara, RO , Romaia acraciu@ieat.ro, gabrielistrate@acm.org Abstract. We prove that the propositioal traslatios of the Keser- Lovász theorem have polyomial size exteded Frege proofs ad quasipolyomial size Frege proofs. We preset a ew coutig-based combiatorial proof of the Keser-Lovász theorem that avoids the topological argumets of prior proofs. We itroduce a miiaturizatio of the octahedral Tucer lemma, called the trucated Tucer lemma. The propositioal traslatios of the trucated Tucer lemma are polyomial size ad they imply the Keser-Lovász priciples with polyomial size Frege proofs. It is ope whether they have (quasi-)polyomial size Frege or exteded Frege proofs. 1 Itroductio This paper discusses proofs of Lovász s theorem about the chromatic umber of Keser graphs, ad the proof complexity of propositioal traslatios of the Keser-Lovász theorem. We give a ew proof of the Keser-Lovász theorem that uses a simple coutig argumet istead of the topological argumets used i prior proofs. Our argumets ca be formalized i propositioal logic to give polyomial size exteded Frege proofs ad quasi-polyomial size Frege proofs. Frege systems are soud ad complete proof systems for propositioal logic with a fiite schema of axioms ad iferece rules. The typical example is a textboo style propositioal proof system usig modus poes as its oly rule of iferece, ad all Frege systems are polyomially equivalet to this system. Exteded Frege systems are Frege systems augmeted with the extesio rule, Supported i part by NSF grats DMS ad CCF Supported i part by grat TIN C4-1. Supported i part by NSF grats DMS ad CCF , ad Simos Foudatio award Supported i part by IDEI grat PN-II-ID-PCE Structure ad computatioal difficulty i combiatorial optimizatio: a iterdiscipliary approach.

2 which allows variables to abbreviate complex formulas. Frege proofs are able to reaso usig Boolea formulas; whereas exteded Frege proofs ca reaso usig Boolea circuits. Sice Boolea formulas are cojectured to require expoetial size to simulate Boolea circuits, it is geerally cojectured that there is a expoetial separatio betwee the sizes of Frege proofs ad exteded Frege proofs. This is oe of the most importat ope questios i proof complexity. However, as discussed by Boet, Buss ad Pitassi [2] ad more recetly by [1, 3], we do ot have ay examples of combiatorial tautologies that are cojectured to expoetially separate Frege proof size from exteded Frege proof size. These prior wors discussed the complexity of propositioal proofs of a umber of combiatorial priciples, icludig the pigeohole priciple ad Fral s theorem. The preset paper exteds this wor by givig short Frege proofs of the propositioal traslatios of the Keser-Lovász theorem. These upper bouds are of iterest, i part because the give a ovel proof of the Keser-Lovász theorem, ad i part because they help us uderstad the questio of whether Frege systems (quasi-)polyomially simulate exteded Frege systems. Istrate ad Crãciu [8] proposed the tautologies based o the Keser-Lovász theorem as a cadidate for a family of tautologies that could expoetially separate Frege ad exteded Frege proof sizes. These tautologies are of iterest because the ow proofs of the Keser-Lovász theorem use (at least implicitly) a topological fixed-poit lemma. The most combiatorial proof is by Matouše[11] ad is ispired by the octahedral Tucer lemma. Ziegler [13] gives other combiatorial proofs, but they also are ispired by topological costructios. Our ew proofs mostly avoid topological argumets ad use a coutig argumet istead. The coutig argumets reduce the geeral case of the Keser- Lovász theorem to small istaces of size 2 4. For fixed, there are oly fiitely may small istaces, ad they ca be verified by exhaustive eumeratio. For fixed values of, these coutig-based proofs ca be coverted ito polyomial size exteded Frege proofs ad quasi-polyomial size Frege proofs. This shows that the Keser-Lovász theorem does ot give a expoetial separatio betwee Frege ad exteded Frege proof size. It is surprisig that the topological argumets ca be largely elimiated from the proof of the Keser-Lovász theorem. The oly remaiig use of topological argumets is to establish the small istaces. It would be iterestig to give a additioal argumet that avoids havig to prove the small istaces separately. Of idepedet iterest, we also preset a miiaturized versio of the octahedral Tucer lemma called the trucated Tucer lemma. Whe the usual Tucer lemma is traslated ito propositioal formulas, the resultig formulas are expoetially large. I cotrast, the trucated Tucer lemma has polyomial size propositioal traslatios. We prove that the Keser-Lovász tautologies have polyomial size costat depth Frege proofs if the propositioal formulas for trucated Tucer lemma are give as additioal hypotheses. It remais ope whether these trucated Tucer lemma priciples have polyomial size Frege or exteded Frege proofs.

3 The (, )-Keser graph is defied to be the udirected graph whose vertices are the -subsets of {1,...,}; there is a edge betwee two vertices iff those vertices have empty itersectio. The Keser-Lovász theorem states that Keser graphs have a large chromatic umber: Theorem 1 (Lovász [10]). Let 2 > 1. The (,)-Keser graph has o colorig with 2 +1 colors. It is easy to show that the (,)-Keser graph has a colorig with 2+2 colors (see Sect. 6), so the boud 2 +1 is optimal. For = 1, the Keser- Lovász theorem is just the pigeohole priciple. Istrate ad Crãciu [8] oted that, for fixed values of, the propositioal traslatios of the Keser-Lovász theorem have polyomial size i. They preseted argumets that ca be formalized by polyomial size Frege proofs for = 2, ad by polyomial size exteded Frege proofs for = 3. This left ope the possibility that the = 3 case could expoetially separate the Frege ad exteded Frege systems. It was also left ope whether the > 3 case of the Keser-Lovász theorem gave tautologies that require expoetial size exteded Frege proofs. As discussed above, the preset paper refutes these possibilities, as the ext two theorems state the existece of polyomial size exteded Frege proofs ad quasi-polyomial size Frege proofs. Cosequetly, the Keser-Lovász theorem is ot a cadidate for expoetially separatig the Frege ad exteded Frege systems. Theorem 2. For fixed parameter 1, propositioal traslatios of the Keser- Lovász theorem have polyomial size exteded Frege proofs. Theorem 3. For fixed parameter 1, propositioal traslatios of the Keser- Lovász theorem have quasi-polyomial size Frege proofs. It would be iterestig to ow if there are short Frege proofs of the Keser- Lovász theorem based o the argumet i Matouše [11]. The obstacle i doig so is that the propositioal traslatios of the octahedral Tucer lemma are of expoetial size i. Sect. 4 itroduces the trucated Tucer lemma, which has a polyomial size traslatio ito propositioal formulas. The trucated Tucer lemma is show to follow from the Tucer lemma, ad the Keser-Lovász theorem follows from the trucated Tucer lemma. It is ope whether the trucated Tucer lemma has short Frege or exteded Frege proofs. 1.1 Prelimiaries Let [] be the set {1,...,}; members of [] are called odes. We idetify ( ) with the set of -subsets of []. The (,)-Keser graph is the udirected graph with vertex set ( ( ) ad with edges {S,T} such that S,T ) ad S T =. Defiitio 4. A m-colorig of the (,)-Keser graph is a map c from ( ) to [m], such that for S,T ( ), if S T =, the c(s) c(t). If l [m] is a color, the the color class P l is the set c 1 ({l}).

4 Defiitio 5. A color class P l is star-shaped if P l is o-empty. If P l is star-shaped, the ay i P l is called a cetral elemet of P l. The ext lemma upper bouds the size of color classes that are ot starshaped. It will be used i our proof of the Keser-Lovász theorem to establish the existece of star-shaped color classes. The idea is that o-star-shaped are too small to cover all ( ) tuples with oly o-star-shaped color classes. Lemma 6. Let c be a m-colorig of ( ). If Pl is ot star-shaped, the ( ) 2 P l 2. 2 Proof. Suppose P l is ot star-shaped. If P l is empty, the claim is trivial. So suppose P l, ad let S 0 = {a 1,...,a } be some elemet of P l. Sice P l is ot star-shaped, there must be sets S 1,...,S P l with a i / S i for i = 1,...,. To specify a arbitrary elemet S of P l, we do the followig. Sice S S 0 is o-empty, we first specify some a i S S 0. As S S i is o-empty, we the specify some a j S S i. By costructio, a i a j, so S is fully specified by the possible values for a i, the possible values for a j, ad the ( 2 2) possible values for the remaiig members of S. Therefore, P l 2( 2 2). Frege systems [7] are textboo style soud ad implicatioally complete propositioal proof systems for tautologies. They have a fiite schema of axioms ad iferece rules. The stadard rule of iferece is modus poes: from A ad A B, ifer B. Exteded Frege systems are Frege systems augmeted with the extesio rule, which allows a formula to be abbreviated by a ew propositioal variable. The size of a Frege or exteded Frege proof is measured by coutig the umber of symbols i the proof [6]. We will ot write out i detail the Frege ad exteded Frege proofs of Theorems 2 ad 3. Istead we will give iformal argumets that are straightforward to traslate ito Frege or exteded Frege proofs usig the techiques i [4]. More bacgroud o propositioal proof systems is give i the surveys [5,9,12]. The formulas Keser are the atural propositioal traslatios of the statemet that there is o ( 2+1)-colorig of the (,)-Keser graph: Defiitio 7. Let 2 > 1, ad m = 2+1.For A ( ) ad i [m], the propositioal variable p A,i has the iteded meaig that vertex A of the Keser graph is assiged the color i. The formula Keser is p A,i (p A,i p B,i ). A ( i [m] ) 2 Mathematical Argumets A,B ( ) A B= i [m] This sectio gives our ew proofs of the Keser-Lovász theorem. Sectio 2.1 gives the first versio, which we later show is formalizable as polyomial size

5 exteded Frege proofs. Sectio 2.2 gives a slightly more complicated but more efficiet proof, which is later show to be formalizable with quasi-polyomial size Frege proofs. 2.1 Argumet for Exteded Frege Proofs This sectio gives our ew proof of the Keser-Lovász theorem. Sect. 3.1 will outlie its formalizatio with polyomial size exteded Frege proofs. Let > 1 be fixed. We prove the Keser-Lovász theorem by iductio o. Thebasecasesfortheiductioare = 2,...,N()whereN()isthecostat depedig o specified i Lemma 8. We shall show that N() is o greater tha 4. Sice is fixed, there are oly fiitely may base cases. Sice the Keser-Lovász theorem is true, these base cases ca all be proved by a fixed Frege proof of fiite size (depedig o ). Therefore, i our proof below, we oly show the iductio step. Lemma 8. Fix > 1. There is a N() so that, for > N(), ay ( 2+1)- colorig of ( ) has at least oe star-shaped color class. Proof. Suppose that a colorig c does ot have ay star-shaped color class. Sice there are 2 +1 may color classes, Lemma 6 implies that ( ) ( ) 2 ( 2 +1) 2. (1) 2 For fixed, the left had side of (1) is Θ( 1 ) ad the right-had side is Θ( ). Thus, there exists a N() such that (1) fails for all > N(). Hece for > N(), there must be at least oe star-shaped color class. We are ow ready to give our first proof of the Keser-Lovász theorem. Proof (of Theorem 1). Fix > 1. By Lemma 8, there is some N() such that for > N(), ay ( 2 + 1)-colorig c of ( ) has a star-shaped color. As discussed above, the cases of N() cases are hadled by exhaustive search ad the truth of the Keser-Lovász theorem. For > N(), we prove the claim by ifiite descet. I other words, we show that if c is a ( 2+1)-colorig of ( ), the there is some c which is a (( 1) 2 +1)-colorig of ( ) 1. By Lemma 8, the colorig c has some star-shaped color class P l with cetral elemet i. Without loss of geerality, i = ad l = Lettig c = c ( ) 1 betherestrictioofctothedomai ( ) 1,itisclearthatc isa(( 1) 2+1)- colorig of ( ) 1. This completes the proof. For completeess, let us calculate a upper boud o the value of N(). Equatio (1) is equivalet to ( 2 +1) 3 ( 1) ( 1). (2) Sice 2 1 1, (2) implies that ( 1) 4 > ( 1) ad thus that < 4. Thus, (1) will be false if 4 ; so N() < 4.

6 2.2 Argumet for Frege Proofs We ow give a secod proof of the Keser-Lovász theorem. The proof above required N() rouds of ifiite descet to trasform a Keser graph o odes to oe o N() odes. Our secod proof replaces this with oly O(log) may rouds, ad this efficiecy will be ey for formalizig this proof with quasipolyomial size Frege proofs i Sect We refie Lemma 8 to show that for sufficietly large, there are may (i.e., a costat fractio) shar-shaped color classes. The idea is to combie the upper boud of Lemma 6 o the size of o-star-shaped color classes with the trivial upper boud of ( 1 1) o the size of star-shaped color classes. Lemma 9. Fix > 1 ad 0 < β < 1. The there exists a N(,β) such that for > N(,β), if c is a ( 2 +1)-colorig of ( ), the c has at least β may star-shaped color classes. The value of N(,β) ca be set equal to 3 ( β) 1 β. The proof of Theorem 3 applies Lemma 9 with β = 1/2. Lemma 9 implies that for > N(,1/2), ay ( 2+1)-colorig of ( ) has at least 2 may star-shaped color classes. Proof (of Lemma 9). Let > N(,β) = 3 ( β) 1 β,adsupposecisa( 2+1)- colorig of ( ). Let α be the umber of star-shaped color classes of c. It is clear that a upper boud o the size of each star-shaped color class is ( 1 1). There are α 2+1 may star-shaped classes, ad Lemma 6 upper bouds their size by 2( 2 2). This implies that ( 1 1 ) ( ) 2 α+ 2 ( α 2 +1) 2 ( ). (3) Assume for a cotradictio that α < β. Sice > 3 ( β) 1 β, 0 < β < 1, ad 2, we have 1 > 3 ( 1) > 2 ( 1). Therefore, ( ) 1 ( 1 > 2 2 2), ad if α is replaced by the larger value β, the left had side of (3) icreases. Thus, ( ) ( ) 1 2 ( 1 β +2 ) ( ) 2 β 2 +1 >. Sice ( ) 1 1 = ( ) ad β β 2 +1 = 2 +1, ) ( ( β We have β 2 ( 2 2 ) 2 +1 β ( 1) > Therefore, > (1 β) 3 ( 1) β ( 1) > (1 β)( 1). Dividig by 1 gives 3 ( β) > (1 β), cotradictig > 3 ( β) 1 β. ).

7 We ow give our secod proof of the Keser-Lovász theorem. Proof (of Theorem 1). Fix > 1. By Lemma 9, if > N(,1/2) ad c is a ( 2 + 1)-colorig of ( ), the c has at least /2 may star-shaped color classes. We prove the Keser-Lovász theorem by iductio o. The base cases are for 2 N(,1/2), ad there are oly fiitely of these, so they ca be exhaustively prove. For > N(,1/2), we structure the iductio proof as a ifiite descet. I other words, we show that if c is a ( 2 + 1)-colorig of ( ), the there is some c that is a (( 2 ) 2 +1)-colorig of ( ) 2. For simplicity of otatio, we assume 2 is a iteger. If this is ot the case, we really mea to roud up to the earest iteger 2. By permutig the color classes ad the odes, we ca assume w.l.o.g. that the 2 color classes P l for l = ,..., 2 +1 are star-shaped, ad each such P l has cetral elemet l+2 1. That is, the last 2 may color classes are star-shaped ad their cetral elemets are the last 2 odes i []. Defie c to be the colorig of ( ) /2 which assigs the same colors to ( 2 1 -tuples as c. The map c is a ( )-colorig of 2 ), sice 2 = This completes the proof of the iductio step. Whe formalizig the above argumet as quasi-polyomial size Frege proof, it will be importat to ow how may iteratios of the procedure are required to reach the base cases, so let us calculate this. 2 1 After s iteratios of this procedure, we have a (( 2 )s 2+1)-colorig ( ( 2 1 ) of 2 )s. We pic s large eough so that ( )s is less tha N(,1/2). I other words, sice is costat, s = log (2 1) = O(log) will suffice, ad oly O(log) may rouds of the procedure are required. We do ot ow if the boud i Lemma 9 is optimal or close to optimal. Appedix 6 discusses the best examples we ow of colorigs with large umbers of o-star-shaped color classes. 3 Formalizatio i Propositioal Logic 3.1 Polyomial Size Exteded Frege Proofs We setch the formalizatio of the argumet i Sect. 2.1 as a polyomial size exteded Frege proof, establishig Theorem 2. We cocetrate o showig how to express cocepts such as star-shaped color class with polyomial size propositioal formulas. We leave to the reader to chec the details of how (exteded) Frege proofs ca prove properties of these cocepts.

8 Fix values for ad with > N(). We describe a exteded Frege proof of Keser. We have variables p S,j (recall Defiitio 7), collectively deoted just p. The proof assumes Keser ( p) is false, ad proceeds by cotradictio. The mai step is to defie ew variables p ad prove that Keser 1 ( p ) fails. This will be repeated util reachig a Keser graph over oly N() odes. For this, let StarShaped(i,l) be a formula that is true whe i [] is a cetral elemet of the color class P l ; amely, StarShaped(i, l) := p S,l. S ( ),i/ S We use StarShaped(l) := i StarShaped(i,l) to express that P l is star-shaped. The exteded Frege proof defies the istace of the Keser-Lovasz priciple Keser 1 by discardig oe ode ad oe color. The first star-shaped color class P l is discarded; accordigly, we let DiscardColor(l) := StarShaped(l) StarShaped(l ). The ode to be discarded is the least cetral elemet of the discarded P l : DiscardNode(i) := [DiscardColor(l) StarShaped(i,l) ] StarShaped(i,l). l After discardig the ode i ad color class P l, the remaiig odes ad colors are reumbered to the rages [ 1] ad [ 2], respectively. I particular, the ew color j (i the istace of Keser 1 ) correspods to the old color j l (i the istace of Keser ) where { j l j if j < l = j +1 if j l. Ad, if S = {i 1,...,i } ( ) 1 is a ew vertex (for the Keser 1 istace), the it correspods to the old vertex S i ( ) (for the istace of Keser ), where S i = {i 1,i 2,...,i } with { i i t if i t < i t = i t +1 if i t i. For each S ( ) 1 ad j [ 1], the exteded Frege proof uses the extesio rule to itroduce a ew variable p S,j defied as follows l <l i <i p S,j i,l ( DiscardNode(i) DiscardColor(l) ps i,j l ). As see i the defiitio by extesio, p S,j is defied by cases, oe for each pair i,l of odes ad colors such that the ode i is the least cetral elemet of

9 the P l color class, where P l is the first star-shaped color class. The exteded Frege proof the shows that Keser ( p) implies Keser 1 ( p ), i.e., that if the variables p S,j defie a colorig, the the variables p S,j also defie a colorig. For this, it is ecessary to show that there is at least oe star-shaped color class; this is provable with a polyomial size exteded Frege proof (eve a Frege proof) usig the costructio of Lemma 8 ad the coutig techiques of [4]. The exteded Frege proof iterates this process of removig oe ode ad oe color util it is show that there is a colorig of ( ) N(). This is the refuted by exhaustively cosiderig all graphs with N() odes. 3.2 Quasi-polyomial Size Frege Proofs This sectio discusses some of the details of the formalizatio of the argumet i Sect. 2.2 as quasi-polyomial size Frege proofs,, establishig Theorem 3. First we will form a exteded Frege proof, the modify it to become a Frege proof. As before, the proof starts with the assumptio that Keser ( p) is false. As we describeext, the extededfregeproofthe itroducesvariables p byextesio so that Keser /2 is false. This process will be repeated O(log) times. The fial Frege proof is obtaied by uwidig the defiitios by extesio. For a set of formulas X, ad for t > 0, let X < t deote a formula that is true whe the umber of true formulas i X is less tha t. If there are polyomially may formulas i X, each of polyomial size, the X < t is of polyomial size usig the costructio i [4]. The formula X = t is defied similarly. The formulas StarShaped(i, l) ad StarShaped(l) are the same as i Sect A color l is ow discarded if it is amog the least /2 star-shaped color classes. DiscardColor(l) := StarShaped(l) ( {StarShaped(l ) : l l} /2) The discarded odes are the least cetral elemets of the discarded color classes. DiscardNode(i) := [DiscardColor(l) StarShaped(i,l) ] StarShaped(i,l). l i <i The remaiig, o-discarded colors ad odes are reumbered to form a istace of Keser /2. For this, the formula ReumNode(i,i) is true whe the ode i is the ith ode that is ot discarded; similarly ReumColor(j,j) is true whe the color j is the jth color that is ot discarded. ReumNode(i,i) := ( { DiscardNode(i ) : i < i } = i 1) DiscardNode(i ) ReumColor(j,j) := ( { DiscardColor(j ) : j < j } = j 1) DiscardColor(j ) For each S = {i 1,...,i } ( ) /2 ad j [( /2) 2+1], we defie by extesio p S,j ( ) (ReumNode(i t,i t )) ReumColor(j,j) p {i 1,...,i. },j i 1,...i,j t=1

10 The Frege proof the argues that if the variables p S,j defie a colorig, the thevariablesp S,j defieacolorig,i.e.,that Keser ( p) Keser /2 ( p ). The mai step for this is provigthere are at least /2 star-shapedcolorclasses byformalizigthe proofoflemma9;thiscabe doewithpolyomialsizefrege proofs usig the coutig techiques from [4]. After that, it is straightforward to prove that, for each S ( ) /2 ad j [( /2) 2+1], the variable p S,j is well-defied; ad that the p collectively falsify Keser /2. This is iterated O(log) times util fewer tha N(,1/2) odes remai. The proof cocludes with a hard-coded proof that there are o such colorigs of small Keser graphs. To form the quasi-polyomial size Frege proof, we uwid the defiitios by extesio. Each defiitio by extesio was polyomial size; they are ested to a depth of O(log ). So the resultig Frege proof is quasi-polyomial size. 4 The Trucated Tucer Lemma This sectio itroduces the trucated Tucer lemma. We show that the usual Tucer lemma implies the trucated Tucer lemma ad the trucated Tucer lemma implies the Keser-Lovász theorem. The trucated Tucer lemma is of particular iterest, sice its propositioal traslatios are oly polyomial size; i cotrast, the propositioal traslatios of the usual Tucer lemma are of expoetial size. Additioally, there are polyomial size costat depth Frege proofs of the Keser-Lovász tautologies from the trucated Tucer tautologies. Our proof of the trucated Tucer lemma from the Tucer lemma borrows techiques ad otatio from Matouše [11]. Defiitio 10. Let 1. The octahedral ball B is: B := {(A,B) : A,B [] ad A B = }. Let 1. The trucated octahedral ball B is: ( ) } B {(A,B) := : A,B { }, A B =, ad (A,B) (, ). Defiitio 11. Let > 1. A mappig λ : B {1,±2,...,±} is atipodal if λ(, ) = 1, ad for all other pairs (A,B) B, λ(a,b) = λ(b,a). Let 2 > 1. A mappig λ : B {±2,...,±} is atipodal if for all (A,B) B, λ(a,b) = λ(b,a). Note that 1 is ot i the rage of λ, ad (, ) is the oly member of B that is mapped to 1 by λ. ForA [], leta deotethe least elemetsofa.by covetio =, but otherwise the otatio is used oly whe A. Defiitio 12. Let be the partial order o sets i ( ) { } defied by A1 A 2 iff (A 1 A 2 ) = A 2.

11 As we will see mometarily i Defiitio 14 ad Lemmas 15 ad 16, the Tucer lemma uses the subset relatio o [], but the trucated Tucerlemma uses istead the stroger partial order o ( ). Lemma 13. The relatio is a partial order with its least elemet. Proof. It is clearly reflexive. For ati-symmetry, A 1 A 2 ad A 2 A 1 imply that A 1 = (A 1 A 2 ) = (A 2 A 1 ) = A 2. For trasitivity: Suppose A 1 A 2 ad A 2 A 3. The (A 1 A 2 ) = A 2 ad (A 2 A 3 ) = A 3. This implies that A 3 = (A 2 A 3 ) = ((A 1 A 2 ) A 3 ) = (A 1 (A 2 A 3 ) ) = (A 1 A 3 ). Therefore A 1 A 3. That is the least elemet is clear from the defiitio. Defiitio 14. Two pairs (A 1,B 1 ) ad (A 2,B 2 ) i B are complemetaryw.r.t. a atipodal map λ o B if A 1 A 2, B 1 B 2 ad λ(a 1,B 1 ) = λ(a 2,B 2 ). For (A 1,B 1 ) ad (A 2,B 2 ) i B, write (A 1,B 1 ) (A 2,B 2 ) whe A 1 A 2, B 1 B 2, ad A i B j = for i,j {1,2}. Ad, (A 1,B 1 ) ad (A 2,B 2 ) are -complemetary w.r.t. a atipodal map λ o B if λ(a 1,B 1 ) = λ(a 2,B 2 ). Lemma 15 (Tucer lemma). If λ : B {1,±2,...,±} is atipodal, the there exists two elemets i B that are complemetary. Lemma 16 (Trucated Tucer). Let 2 > 1. If λ : B {±2...,±} is atipodal, the there are two elemets i B that are -complemetary. For a proof of Lemma 15, see [11]. Appedix 5 proves Lemma 15 implies Lemma 16, hece the trucated Tucer lemma is true. The trucated Tucer lemma has polyomial size propositioal traslatios. For each (A,B) B, ad for each i {±2,...,±}, let p A,B,i be a propositioal variable with the iteded meaig that p A,B,i is true whe λ(a,b) = i. The followig formula At( p) states that the map is total ad atipodal: (p A,B,i p B,A, i ). (A,B) B i {±2,...,±} The followig formula Comp( p) states that there exists two elemets i B that are -complemetary: (p A1,B 1,i p A2,B 2, i). (A 1,B 1),(A 2,B 2) B, (A 1,B 1) (A 2,B 2) i {±2,...,±} The trucated Tucer tautologies are defied to be At( p) Comp( p). (We could add a additioalhypothesis, that for eacha,b there is at most oe isuch that p A,B,i, but this is ot eeded for the Tucer tautologies to be valid.) There are < 2 members (A,B) i B. Hece, for fixed, there are oly polyomially may variables p A,B,i, ad the trucated Tucer tautologies have size polyomially bouded by. O the other had, the propositioal traslatio of the usual Tucer lemma requires a expoetial umber of propositioal variables i, sice the cardiality of B is expoetial i.

12 Proof (Theorem 1 from the trucated Tucer lemma). Let c : ( ) {2,...,} be a colorig of ( ). We show that this implies the existece of a atipodal map ) λ o B that has o -complemetary pairs. Let be a total order o { } that refies the partial order. Defie λ(a,b) to be c(a) if A > B ( ad c(b) if B > A. We argue that there are o -complemetary pairs i B with respect to λ. Suppose there are, say (A 1,B 1 ) ad (A 2,B 2 ). Sice λ must assig these opposite sigs, either A 1 < B 1 B 2 < A 2 or B 1 < A 1 A 2 < B 2. I the former case it must be that, c(b 1 ) = c(a 2 ) ad i the latter case that c(a 1 ) = c(b 2 ). Sice B 1 A 2 ad A 1 B 2 are empty i either case we have a cotradictio, sice c was assumed to be a colorig. The above proof of the Keser-Lovász theorem from the trucated Tucer lemma ca be readily traslated ito polyomial size costat depth Frege proofs. Questio 17. Do the propositioal traslatios of the Trucated Tucer lemma have short (exteded) Frege proofs? Refereces 1. Aiseberg, J., Boet, M.L., Buss, S.: Quasi-polyomial size Frege proofs of Fral s theorem o the trace of fiite sets (201?), to appear i Joural of Symbolic Logic 2. Boet, M.L., Buss, S.R., Pitassi, T.: Are there hard examples for Frege systems? I: Clote, P., Remmel, J. (eds.) Feasible Mathematics II. pp Birhäuser, Bosto (1995) 3. Buss, S.: Quasipolyomial size proofs of the propositioal pigeohole priciple (2014), submitted for publicatio 4. Buss, S.R.: Polyomial size proofs of the propositioal pigeohole priciple. Joural of Symbolic Logic 52, (1987) 5. Buss, S.R.: Propositioal proof complexity: A itroductio. I: Berger, U., Schwichteberg, H. (eds.) Computatioal Logic, pp Spriger-Verlag, Berli (1999) 6. Coo, S.A., Rechow, R.A.: O the legths of proofs i the propositioal calculus, prelimiary versio. I: Proceedigs of the Sixth Aual ACM Symposium o the Theory of Computig. pp (1974) 7. Coo, S.A., Rechow, R.A.: The relative efficiecy of propositioal proof systems. Joural of Symbolic Logic 44, (1979) 8. Istrate, G., Crãciu, A.: Proof complexity ad the Keser-Lovász theorem. I: Theory ad Applicatios of Satisfiability Testig (SAT). pp Lecture Notes i Computer Sciece 8561, Spriger Verlag (2014) 9. Krajíče, J.: Bouded Arithmetic, Propositioal Calculus ad Complexity Theory. Cambridge Uiversity Press, Heidelberg (1995) 10. Lovász, L.: Keser s cojecture, chromatic umber, ad homotopy. Joural of Combiatorial Theory, Series A 25(3), (1978) 11. Matouše, J.: A combiatorial proof of Keser s cojecture. Combiatorica 24(1), (2004) 12. Segerlid, N.: The complexity of propositioal proofs. Bulleti of Symbolic Logic 13(4), (2007) 13. Ziegler, G.M.: Geeralized Keser colorig theorems with combiatorial proofs. Ivetioes Mathematicae 147(3), (2002)

13 5 Appedix: Proof of the Trucated Tucer Lemma We prove the trucated Tucer lemma from the Tucer lemma: Proof (of Lemma 16 from Lemma 15). We show the cotrapositive. Suppose Lemma16isfalse.Iotherwords,thereissomeatipodalλ : B {±2,...,±} ad there are o -complemetary pairs of elemets i B with respect to λ. We will defie a atipodal map λ : B {1,±2,...,±} that has o complemetary pairs of elemets i B with respect to λ, i violatio of Lemma 15. Let be a total order o the subsets of [] that respects cardialities ad refies o elemets i ( ) { }. I other words, if A < B or if A B the A B. Similarly to a costructio i [11], λ (A,B) is defied by cases: Case I: If A < ad B <, the λ (A,B) = { 1+ A + B if A B (1+ A + B ) if B > A. Case II: If A = ad B <, the λ (A,B) = λ(a, ). Similarly, if A < ad B =, the λ (A,B) = λ(,b). Case III: If A ad B, the λ (A,B) = λ(a,b ). The map λ is clearly atipodal sice λ is. Let (A 1,B 1 ) ad (A 2,B 2 ) be members of B with A 1 A 2, B 1 B 2. We wish to show they are ot a complemetary pair for λ, amely that λ (A 1,B 1 ) λ (A 2,B 2 ). The argumet splits ito cases. 1. λ (A 1,B 1 ) ad λ (A 2,B 2 ) are assiged by Case I. If A 1 = A 2 ad B 1 = B 2, the clearly λ (A 1,B 1 ) λ (A 2,B 2 ). Otherwise, at least oe of the iclusios A 1 A 2 ad B 1 B 2 is proper, so A 1 + B 1 < A 2 + B 2. But λ (A 1,B 1 ) = 1 + A 1 + B 1 ad λ (A 2,B 2 ) = 1 + A 2 + B 2, so λ (A 1,B 1 ) λ (A 2,B 2 ). 2. λ (A 1,B 1 ) ad λ (A 2,B 2 ) areassigedby CaseII. Without lossofgeerality, A 1 = ad B 1 <.Sothe A 2 = A 1 ad B 2 <.But theλ (A 1,B 1 ) = λ(a 1, ) ad λ (A 2,B 2 ) = λ(a 1, ), so λ (A 1,B 1 ) λ (A 2,B 2 ). 3. λ (A 1,B 1 ) ad λ (A 2,B 2 ) are both assiged by Case III. It is clear that A i, B j, = for i,j {1,2}, sice A 2 B 2 =. We claim that A 1, A 2, ad B 1, B 2,. This is because A 1 A 2, so (A 1, A 2, ) = (A 1 A 2 ) = A 2,. Therefore A 1, A 2,. The same argumet shows that B 1, B 2,. Sice there are o -complemetary pairs with respect to λ, it must be that λ(a 1,,B 1, ) λ(a 2,,B 2, ). Also, λ (A 1,B 1 ) = λ(a 1,,B 1, ) ad λ (A 2,B 2 ) = λ(a 2,,B 2, ). Hece λ (A 1,B 1 ) λ (A 2,B 2 ).

14 4. λ (A 1,B 1 ), ad λ (A 2,B 2 ) are assiged by Cases II ad III. If this happes, it must be that (A 1,B 1 ) is assiged as i Case II, ad (A 2,B 2 ) is assiged as i Case III. Without loss of geerality, say A 1 =, ad B 1 <. We show that (A 1, ) (A 2,B 2, ). By the same argumet as the previous case, A 1, A 2,. As remared earlier, B 2,, sice the empty set is the least elemet i the partial order. Also, the four sets A 1,, A 1, B 2,, B 2,, ad A 2, B 2, are empty. Sice there are o -complemetary pairs with respect to λ, it must be that λ(a 1,, ) λ(a 2,,B 2, ). By defiitio, λ (A 1,B 1 ) = λ(a 1,, ) ad λ (A 2,B 2 ) = λ(a 2,,B 2, ). Hece λ (A 1,B 1 ) λ (A 2,B 2 ). 5. The oly remaiig case is whe oe of λ (A 1,B 1 ) ad λ (A 2,B 2 ) is assiged by Case I, ad the other is assiged by Case II or III. It must be that λ (A 1,B 1 ) is assiged by Case I ad λ (A 2,B 2 ) by Case II or III. Observe that λ (A 1,B 1 ) = 1 + A 1 + B 1 2 1, ad that λ (A 2,B 2 ) 2. Therefore λ (A 1,B 1 ) λ (A 2,B 2 ). This establishes that λ is a atipodal map with o complemetary pairs, hece Lemma 15 is false. This completes the proof of the cotrapositive. 6 Appedix: Optimal Colorigs of Keser Graphs It is well-ow that ( ) has a ( 2+2)-colorig[10]. A simple costructio of such a colorig, which we call c 1, is as follows. For S ( ), defie c1 (S) by: (1) If S [2 1], let c 1 (S) = max(s) (2 2). Clearly 1 < c 1 (S) 2+2. (2) Otherwise, let c 1 (S) = 1. We claim that c 1 defies a proper colorig. By costructio, if c 1 (S) > 1, the c 1 (S) + (2 2) S. Thus, if c 1 (S) = c 1 (S ) > 1, the S S ad S ad S are ot joied by a edge i the Keser graph. O the other had, if c 1 (S) = 1, the S cotais elemets from the set [2 1]. Ay two such subsets have oempty itersectio, ad therefore if c 1 (S) = c 1 (S ) = 1, the agai S S. Note that c 1 cotais 2+1 may star-shaped colorclasses, ad oly oe o-star-shaped color class. I view of Lemma 9, it is iterestig to as whether it is possible to give ( 2 + 2)-colorigs with fewer star-shaped color classes ad more o-starshaped color classes. The ext theorem gives the best costructio we ow. Theorem 18. Let 1 ad There is a ( 2+2) colorig c 1 of ( ) which has 1 may o-star-shaped color classes ad oly 3 +3 may star-shaped color classes. Proof. To costruct c 1, partitio the set [] ito may subsets T 1,...,T 2+2 as follows. For i 3+3, T i is chose to be a sigleto set, say T i = { i+1}. The remaiig 1 may T i s are subsets of size 3, say T i = {j 2,j 1,j} where j = i ( 3+3). Sice = ( 3+3)+3( 1),

15 the sets T i partitio [], ad each T i has cardiality either 1 or 3. For S a subset of of cardiality, defie the color c 1 (S) to equal the least i such that S T i > 1 2 T i. We claim there must exist such a i. If ot, the S cotais o members of the sigleto subsets T i ad at most oe member of each of the subsets T i of size three. But there are oly 1 may subsets of size three, cotradictig S =. It is easy to chec that if c 1 (S) = c 1 (S ) the S S. Thus c 1 is a colorig. Furthermore, c 1 has 1 may o-star-shaped classes ad 3 +3 may star-shaped classes. Theorem 18 ca be exteded to show that whe < 3 + 3, there is a 2+2 colorig with o star-shaped color class. The proof costructio uses a similar idea, based o the fact that [] ca be partitioed ito 2+2 may subsets, each of odd cardiality 3. We leave the details to the reader. Questio 19. Do there exist ( 2 + 2)-colorigs of the (,)-Keser graphs with more tha 1 may o-star-shaped color classes?