Benny Sudakov

Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors

A distribution <i>X</i> over binary strings of length <i>n</i> has min-entropy <i>k</i> if every string has probability at most 2^-<i>k</i> in <i>X</i>. We say that <i>X</i> is a δ-source if its rate <i>k</i>⁄<i>n</i> is at least δ.We give the following new explicit instructions (namely, poly(n)- time computable functions) of <i>deterministic</i>extractors, dispersers and related objects. All work for any fixed rate δ>0. No previous explicit construction was known for either of these, for any δ‹1⁄2. The first two constitute major progress to very long-standing open problems. <ol><li><b>Bipartite Ramsey</b> <i>f</i><inf>1</inf>: (0,1)ⁿ)² →0,1, such that for any two independent δ-sources <i>X</i><inf>1</inf>, <i>X</i><inf>2</inf> we have <i>f</i><inf>1</inf>(<i>X</i><inf>1</inf>,<i>X</i><inf>2</inf>) = 0,1 This implies a new explicit construction of <i>2N</i>-vertex bipartite graphs where no induced <i>N</i>^δ by <i>N</i>^δ subgraph is complete or empty.</li> <li><b>Multiple source extraction</b> <i>f</i><inf>2</inf>: (0,1<i>n</i>)³→0,1 such that for any three independent δ-sources <i>X</i><inf>1</inf>,<i>X</i><inf>2</inf>,<i>X</i><inf>3</inf> we have that <i>f</i><inf>2</inf>(<i>X</i><inf>1</inf>,<i>X</i><inf>2</inf>,<i>X</i><inf>3</inf>) is (<i>o</i>(1)-close to being) an unbiased random bit.</li> <li><b>Constant seed condenser</b>² <i>f</i><inf>3</inf>: <i>ⁿ</i> →(0,1^<i>m</i>)^<i>c</i>, such that for any δ-source <i>X</i>, one of the <i>c</i> output distributions <i>f</i><inf>3</inf>(<i>X</i>)<inf><i>i</i></inf>, is a 0.9-source over 0,1<i>^m</i>. Here <i>c</i> is a constant depending only on δ.</li><li><b>Subspace Ramsey</b> <i>f</i>4: 0,1<i>ⁿ</i>→0,1 such that for any <i>affine</i>-δ-source³ <i>X</i> we have <i>f</i><inf>4</inf>(<i>X</i>)= 0,1.</li></ol>The constructions are quite involved and use as building blocks other new and known gadgets. But we can point out two important themes which recur in these constructions. One is that gadgets which were designed to work with independent inputs, sometimes perform well enough with correlated, high entropy inputs. The second is using the input to (introspectively) find high entropy regions within itself.

Pseudo-random Graphs

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in proving an enormous number of combinatorial statements, making their role quite hard to overestimate. Their tremendous success serves as a natural motivation for the following very general and deep informal questions: what are the essential properties of random graphs? How can one tell when a given graph behaves like a random graph? How to create deterministically graphs that look random-like? This leads us to a concept of pseudo-random graphs.

The Largest Eigenvalue of Sparse Random Graphs

We prove that, for all values of the edge probability

Coloring Graphs with Sparse Neighborhoods

p(n)

Density theorems for bipartite graphs and related Ramsey-type results

, the largest eigenvalue of the random graph

The Turán Number Of The Fano Plane

G(n, p)

Learning a Hidden Matching

satisfies almost surely

Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels

\lambda_1(G)=(1+o(1))\max\{\sqrt{\Delta}, np\}

Hypergraph Ramsey numbers

, where Δ is the maximum degree of

Bipartite Subgraphs and the Smallest Eigenvalue

G