The Probabilistic Method

The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.

[1]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[2]  P. Turán.  On a Theorem of Hardy and Ramanujan , 1934 .

[3]  P. Erdös,et al.  The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions , 1940 .

[4]  P. Erdös,et al.  On the structure of linear graphs , 1946 .

[5]  P. Erdös.  Some remarks on the theory of graphs , 1947 .

[6]  H. Whitney,et al.  An inequality related to the isoperimetric inequality , 1949 .

[7]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[8]  J. G. Wendel.  A Problem in Geometric Probability. , 1962 .

[9]  J. Moon,et al.  On Sets of Consistent Arcs in a Tournament , 1965, Canadian Mathematical Bulletin.

[10]  L. H. Harper.  Optimal numberings and isoperimetric problems on graphs , 1966 .

[11]  D. Kleitman.  Families of Non-disjoint subsets* , 1966 .

[12]  J. Marica,et al.  Differences of Sets and A Problem of Graham , 1969, Canadian Mathematical Bulletin.

[13]  C. Fortuin,et al.  Correlation inequalities on some partially ordered sets , 1971 .

[14]  R. Graham,et al.  A Constructive Solution to a Tournament Problem , 1971, Canadian Mathematical Bulletin.

[15]  Norbert Sauer,et al.  On the Density of Families of Sets , 1972, J. Comb. Theory A.

[16]  M. Pinsker,et al.  On the complexity of a concentrator , 1973 .

[17]  Paul Erdös,et al.  On a Combinatorial Game , 1973, J. Comb. Theory A.

[18]  A. Joffe.  On a Set of Almost Deterministic $k$-Independent Random Variables , 1974 .

[19]  Wolfgang J. Paul.  A 2.5 n-lower bound on the combinational complexity of Boolean functions , 1975, STOC '75.

[20]  Ronald Fagin,et al.  Probabilities on finite models , 1976, Journal of Symbolic Logic.

[21]  Joel H. Spencer,et al.  Asymptotic lower bounds for Ramsey functions , 1977, Discret. Math..

[22]  Edward M. Wright,et al.  The number of connected sparsely edged graphs , 1977, J. Graph Theory.

[23]  R. Dudley.  Central Limit Theorems for Empirical Measures , 1978 .

[24]  D. E. Daykin,et al.  An inequality for the weights of two families of sets, their unions and intersections , 1978 .

[25]  Alexander Schrijver,et al.  A Short Proof of Minc's Conjecture , 1978, J. Comb. Theory, Ser. A.

[26]  J. Beck,et al.  On 3-chromatic hypergraphs , 1978, Discret. Math..

[27]  János Komlós,et al.  A Note on Ramsey Numbers , 1980, J. Comb. Theory, Ser. A.

[28]  Frank Harary,et al.  Covering and packing in graphs IV: Linear arboricity , 1981, Networks.

[29]  E. Szemerédi,et al.  A Lower Bound for Heilbronn'S Problem , 1982 .

[30]  Wenceslas Fernandez de la Vega,et al.  On the maximum cardinality of a consistent set of arcs in a random tournament , 1983, J. Comb. Theory, Ser. B.

[31]  P. Erdös,et al.  The Greatest Angle Among n Points in the d- Dimensional Euclidean Space , 1983 .

[32]  James B. Shearer,et al.  A note on the independence number of triangle-free graphs , 1983, Discret. Math..

[33]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[34]  Vojtech Rödl,et al.  Geometrical realization of set systems and probabilistic communication complexity , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[35]  Noga Alon,et al.  The maximum number of disjoint pairs in a family of subsets , 1985, Graphs Comb..

[36]  James B. Shearer,et al.  On a problem of spencer , 1985, Comb..

[37]  Vojtech Rödl,et al.  On a Packing and Covering Problem , 1985, Eur. J. Comb..

[38]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .

[39]  Alexander Lubotzky,et al.  Explicit expanders and the Ramanujan conjectures , 1986, STOC '86.

[40]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[41]  László Lovász,et al.  Discrepancy of Set-systems and Matrices , 1986, Eur. J. Comb..

[42]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[43]  P. Raghavan.  Probabilistic construction of deterministic algorithms: Approximating packing integer programs , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[44]  Noga Alon,et al.  Eigenvalues, geometric expanders, sorting in rounds, and ramsey theory , 1986, Comb..

[45]  Fan Chung Graham,et al.  Some intersection theorems for ordered sets and graphs , 1986, J. Comb. Theory, Ser. A.

[46]  Janos Simon,et al.  Probabilistic Communication Complexity , 1986, J. Comput. Syst. Sci..

[47]  Joel H. Spencer,et al.  Sharp concentration of the chromatic number on random graphsGn, p , 1987, Comb..

[48]  Noga Alon,et al.  The monotone circuit complexity of boolean functions , 1987, Comb..

[49]  G. Pisier.  ASYMPTOTIC THEORY OF FINITE DIMENSIONAL NORMED SPACES (Lecture Notes in Mathematics 1200) , 1987 .

[50]  János Komlós,et al.  Deterministic simulation in LOGSPACE , 1987, STOC.

[51]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

[52]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[53]  Noga Alon,et al.  Explicit construction of linear sized tolerant networks , 1988, Discret. Math..

[54]  J. Pach,et al.  Decomposition problems for multiple coverings with unit balls, manuscript. , 1988 .

[55]  A. Meir,et al.  On empty triangles determined by points in the plane , 1988 .

[56]  Vojtech Rödl,et al.  The number of submatrices of a given type in a Hadamard matrix and related results , 1988, J. Comb. Theory, Ser. B.

[57]  Béla Bollobás,et al.  The chromatic number of random graphs , 1988, Comb..

[58]  J. Spencer,et al.  Zero-one laws for sparse random graphs , 1988 .

[59]  Zoltán Füredi,et al.  Matchings and covers in hypergraphs , 1988, Graphs Comb..

[60]  Avi Wigderson,et al.  Monotone circuits for connectivity require super-logarithmic depth , 1990, STOC '88.

[61]  Avi Wigderson,et al.  Dispersers, deterministic amplification, and weak random sources , 1989, 30th Annual Symposium on Foundations of Computer Science.

[62]  Joel H. Spencer,et al.  A useful elementary correlation inequality , 1989, J. Comb. Theory, Ser. A.

[63]  Bernard Chazelle,et al.  Quasi-optimal range searching in spaces of finite VC-dimension , 1989, Discret. Comput. Geom..

[64]  Joel H. Spencer,et al.  Asymptotic behavior of the chromatic index for hypergraphs , 1989, J. Comb. Theory, Ser. A.

[65]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[66]  Noga Alon.  The maximum number of Hamiltonian paths in tournaments , 1990, Comb..

[67]  Joel H. Spencer,et al.  Threshold functions for extension statements , 1990, J. Comb. Theory, Ser. A.

[68]  Gerhard J. Woeginger,et al.  Some new bounds for Epsilon-nets , 1990, SCG '90.

[69]  Tomasz Luczak,et al.  Component Behavior Near the Critical Point of the Random Graph Process , 1990, Random Struct. Algorithms.

[70]  Stephen Suen,et al.  A correlation inequality and a Poisson limit theorem for nonoverlapping balanced subgraphs of a random graph , 1990, Random Struct. Algorithms.

[71]  Richard M. Karp,et al.  The Transitive Closure of a Random Digraph , 1990, Random Struct. Algorithms.

[72]  Joel H. Spencer,et al.  Counting extensions , 1990, J. Comb. Theory, Ser. A.

[73]  Noga Alon,et al.  Transversal numbers of uniform hypergraphs , 1990, Graphs Comb..

[74]  Richard M. Karp,et al.  Parallel Algorithms for Shared-Memory Machines , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[75]  Svante Janson,et al.  Poisson Approximation for Large Deviations , 1990, Random Struct. Algorithms.

[76]  Noga Alon,et al.  A Parallel Algorithmic Version of the Local Lemma , 1991, Random Struct. Algorithms.

[77]  Tomasz Luczak.  A note on the sharp concentration of the chromatic number of random graphs , 1991, Comb..

[78]  József Beck,et al.  An Algorithmic Approach to the Lovász Local Lemma. I , 1991, Random Struct. Algorithms.

[79]  Noga Alon,et al.  On the second eigenvalue of a graph , 1991, Discret. Math..

[80]  Paul Erdös,et al.  Lopsided Lovász Local Lemma and Latin transversals , 1991, Discret. Appl. Math..

[81]  J. Komlos,et al.  Almost tight bounds for $\epsilon$-nets , 1992 .

[82]  P. Valtr.  On the minimum number of empty polygons in planar point sets , 1992 .

[83]  Peter C. Fishburn.  Correlation in Partially Ordered Sets , 1992, Discret. Appl. Math..

[84]  Svante Janson,et al.  The Birth of the Giant Component , 1993, Random Struct. Algorithms.

[85]  Noga Alon,et al.  Explicit Ramsey graphs and orthonormal labelings , 1994, Electron. J. Comb..

[86]  Mihail N. Kolountzakis,et al.  An effective additive basis for the integers , 1995, SODA '94.