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\textbf{Oral Qualifying Exam Syllabus}
\\Mike Neiman
\\Committee: Profs. J. Beck, J. Kahn, D. Ocone, M. Saks
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{\large \textbf{1 Combinatorics}} \\ 

\noindent
\textbf{Basic Enumeration:}
counting arguments, generating functions, binomial coefficients,
recurrence relations, inclusion-exclusion principle, Stirling's
rule%formula 
\\
\textbf{Set Systems:} Sperner's theorem, Erd\H os-Ko-Rado,
Kruskal-Katona, Frankl-Wilson, Harper, Fisher's and generalized
Fisher's inequalities, projective geometries, Baranyai
\\
\textbf{Lattices and Posets:} Dilworth, distributive lattices,
Birkhoff representation theorem, Dilworth's theorem on dimension of
distributive lattices, geometric lattices, M\"obius inversion,
Weisner, Dowling-Wilson 
\\
\textbf{Correlation Inequalities:}
Harris-Kleitman, Fortuin-Kasteleyn-Ginibre, \\
Ahlswede-Daykin, application to Shepp's theorems 
\\
\textbf{Ramsey Theory:} Ramsey's
theorem, infinite Ramsey, K\"onig's lemma, probabilistic lower bounds,
van der Waerden, Hales-Jewett, statement of Szemer\'edi's theorem
\\
\textbf{Discrepancy:} Beck-Fiala, Roth's $\frac{1}{4}$-theorem on
discrepancy of arithmetic progressions \\ \\
{\large \textbf{2 Graph Theory}} 
\\ \\
\textbf{Matching:} K\"onig, Hall, Tutte's 1-factor
theorem, matching polytope 
\\
\textbf{Connectivity:} greedy algorithm
for minimum weight spanning tree, structure of 2-connected graphs,
Menger, Max-Flow-Min-Cut 
\\
\textbf{Planarity:} Euler's formula,
Kuratowski's theorem 
\\
\textbf{Coloring:} 5 color theorem, Brooks,
Vizing, Thomassen's 5-list-coloring of planar graphs, Galvin's proof
of Dinitz conjecture, Lov\'asz's proof of weak perfect graph
conjecture 
\\
\textbf{Extremal Problems:} Tur\'an, statement of
regularity lemma, Erd\H os-Stone, Chv\'atal-R\"odl-Szemer\'edi-Trotter
\\ \\*
{\large \textbf{3 Probabilistic Methods}} 
\\ \\
\textbf{Basics:}
linearity of expectation, alterations, Bonferroni inequalities,
Chebyschev's inequality, Chernoff bound 
\\
\textbf{Second Moment
Method:} general procedure, application to threshold function for
having a certain graph as a subgraph 
\\
\textbf{Lov\'asz Local Lemma:}
symmetric and general versions, applications to hypergraph discrepancy
and Latin transversals 
\\
\textbf{Martingales:} Azuma's inequality,
edge and vertex exposure, application to chromatic number
\\
\textbf{Poisson Paradigm:} Janson inequalities, application to
number of triangles in $G_{n,p}$, Brun's sieve, application to number
of isolated vertices in $G_{n,p}$ 
\\
\textbf{Random Graphs:} monotone
properties, existence of threshold functions, Bollob\'as-Thomason,
probabilistic refutation of Haj\'os conjecture 
\\
\textbf{Entropy:}
basic properties, Shearer's lemma, application to Minc conjecture 
\\
\\
{\large \textbf{4 Probability Theory}} 
\\ \\
\textbf{Probability
Spaces and Random Variables:} algebras and $\sigma$-algebras,
probability spaces, monotone class theorem, independence and product
spaces, random variables, distribution functions, expectation,
independence of random variables, convergence concepts for random
variables, Kolmogorov's 0-1 law 
\\
\textbf{Large Number Laws:} weak law
of large numbers, Borel-Cantelli lemma, strong law of large numbers,
Kolmogorov's three series theorem 
\\
\textbf{Stationary Processes:}
stationarity, measure-preserving transformations, Birkhoff's ergodic
theorem, ergodic theorem for stationary processes 
\\
\textbf{Central
Limit Theorem:} De Moivre-Laplace theorem, weak convergence and
convergence in distribution, characteristic functions, continuity
theorem, Lindeberg-Feller central limit theorem
\\
\textbf{Martingales:} conditional expectation, definition of
(sub)(super) martingales, stopping times, optional stopping theorems,
application to random walks, Doob's martingale inequalities, Doob's
upcrossing inequality, uniform integrability, martingale convergence
theorems, L\'evy's upward convergence theorem
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