Short Proofs of the Kneser-Lovász Coloring Principle

We prove that the propositional translations of the Kneser-Lovasz theorem have polynomial size extended Frege proofs and quasi-polynomial size Frege proofs. We present a new counting-based combinatorial proof of the Kneser-Lovasz theorem that avoids the topological arguments of prior proofs for all but finitely many cases for eachi¾?k. We introduce a miniaturization of the octahedral Tucker lemma, called the truncated Tucker lemma: it is open whether its propositional translations have quasi-polynomial size Frege or extended Frege proofs.

[1]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[2]  Samuel R. Buss,et al.  QUASIPOLYNOMIAL SIZE FREGE PROOFS OF FRANKL’S THEOREM ON THE TRACE OF SETS , 2016, The Journal of Symbolic Logic.

[3]  Eth Zentrum,et al.  A Combinatorial Proof of Kneser's Conjecture , 2022 .

[4]  R. Statman.  Complexity of Derivations from Quantifier-Free Horn Formulae, Mechanical Introduction of Explicit Definitions, and Refinement of Completeness Theorems , 1977 .

[5]  Phuong Nguyen,et al.  The provably total NP search problems of weak second order bounded arithmetic , 2011, Ann. Pure Appl. Log..

[6]  外史 竹内.  Bounded Arithmetic と計算量の根本問題 , 1996 .

[7]  Stephen A. Cook,et al.  Feasibly constructive proofs and the propositional calculus (Preliminary Version) , 1975, STOC.

[8]  Samuel R. Buss,et al.  Propositional proofs and reductions between NP search problems , 2012, Ann. Pure Appl. Log..

[9]  Zoltán Füredi,et al.  Non-trivial intersecting families , 1986, J. Comb. Theory, Ser. A.

[10]  Emil Jerábek,et al.  Dual weak pigeonhole principle, Boolean complexity, and derandomization , 2004, Annals of Pure and Applied Logic.

[11]  R. R. Eilers,et al.  Polynomial Size Proofs for the Propositional Pigeonhole Principle , 2014 .

[12]  G. Ziegler,et al.  Generalized Kneser coloring theorems with combinatorial proofs , 2001, math/0103146.

[13]  László Lovász,et al.  Kneser's Conjecture, Chromatic Number, and Homotopy , 1978, J. Comb. Theory A.

[14]  Stephen A. Cook,et al.  The Relative Efficiency of Propositional Proof Systems , 1979, Journal of Symbolic Logic.

[16]  Peter Dybjer,et al.  Induction-recursion and initial algebras , 2003, Ann. Pure Appl. Log..

[17]  Jan Krajícek,et al.  Bounded arithmetic, propositional logic, and complexity theory , 1995, Encyclopedia of mathematics and its applications.

[18]  Robert M. Freund,et al.  Variable Dimension Complexes Part II: A Unified Approach to Some Combinatorial Lemmas in Topology , 2015, Math. Oper. Res..

[19]  E. Weisstein.  Kneser's Conjecture , 2002 .

[20]  Samuel R. Buss,et al.  Are there Hard Examples for Frege Systems , 1995 .

[21]  Samuel R. Buss,et al.  Annals of Pure and Applied Logic , 2022 .

[22]  Iddo Tzameret,et al.  Short proofs for the determinant identities , 2012, STOC '12.

[23]  Samuel R. Buss,et al.  Improved witnessing and local improvement principles for second-order bounded arithmetic , 2014, ACM Trans. Comput. Log..

[24]  P. Erdös,et al.  INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS , 1961 .

[25]  Samuel R. Buss,et al.  Propositional Proof Complexity An Introduction , 1999 .

[26]  A. J. W. Hilton,et al.  SOME INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS , 1967 .

[27]  Michael J. Todd,et al.  A Constructive Proof of Tucker's Combinatorial Lemma , 1981, J. Comb. Theory, Ser. A.

[28]  P. Pudlák.  On the length of proofs of finitistic consistency statements in first order theories , 1986 .

[29]  Samuel R. Buss,et al.  Propositional Consistency Proofs , 1991, Ann. Pure Appl. Log..

[30]  Gabriel Istrate,et al.  Proof Complexity and the Kneser-Lovász Theorem , 2014, SAT.

[31]  Stephen Cook,et al.  Corrections for "On the lengths of proofs in the propositional calculus preliminary version" , 1974, SIGA.

[32]  Jeremy Avigad.  Plausibly hard combinatorial tautologies. , 1996 .

[33]  Alexander A. Razborov,et al.  Complexity of Propositional Proofs , 2010, CSR.

[34]  Samuel R. Buss,et al.  Polynomial size proofs of the propositional pigeonhole principle , 1987, Journal of Symbolic Logic.

[35]  Robert M. Freund,et al.  Variable Dimension Complexes Part I: Basic Theory , 2015, Math. Oper. Res..

[36]  Peter Frankl,et al.  On the Trace of Finite Sets , 1983, J. Comb. Theory, Ser. A.