Optimistic rollups rely on fraud proofs — interactive protocols executed on Ethereum to resolve conflicting claims about the rollup's state — to scale Ethereum securely.

Abstract. The canonical Ramsey theorem of Erdős and Rado implies that for any graph [Formula: see text], any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph [Formula: see text] contains a monochromatic, lexicographic, or rainbow copy of [Formula: see text]. The least such [Formula: see text] is called the Erdős–Rado number of [Formula: see text], denoted by [Formula: see text]. Erdős–Rado numbers of cliques have received considerable attention, and in this paper we extend this line of research by studying Erdős–Rado numbers of sparse graphs. For example, we prove that if [Formula: see text] has bounded degree, then [Formula: see text] is polynomial in [Formula: see text] if [Formula: see text] is bipartite but exponential in general. We also study the closely related problem of constrained Ramsey numbers. For a given tree [Formula: see text] and given path [Formula: see text], we study the minimum [Formula: see text] such that every edge-coloring of [Formula: see text] contains a monochromatic copy of [Formula: see text] or a rainbow copy of [Formula: see text]. We prove a nearly optimal upper bound for this problem, which differs from the best known lower bound by a function of inverse Ackermann type.

Abstract Given positive integers $$k\le d$$ k ≤ d and a finite field $$\mathbb {F}$$ F , a set $$S\subset \mathbb {F}^{d}$$ S ⊂ F d is (k, c)-subspace evasive if every k-dimensional affine subspace contains at most c elements of S. By a simple averaging argument, the maximum size of a (k, c)-subspace evasive set is at most $$c |\mathbb {F}|^{d-k}$$ c | F | d - k . When k and d are fixed, and c is sufficiently large, the matching lower bound $$\Omega (|\mathbb {F}|^{d-k})$$ Ω ( | F | d - k ) is proved by Dvir and Lovett. We provide an alternative proof of this result using the random algebraic method. We also prove sharp upper bounds on the size of (k, c)-evasive sets in case d is large, extending results of Ben-Aroya and Shinkar. The existence of optimal evasive sets has several interesting consequences in combinatorial geometry. We show that the minimum number of k-dimensional linear hyperplanes needed to cover the grid $$[n]^{d}\subset \mathbb {R}^{d}$$ [ n ] d ⊂ R d is $$\Omega _{d}\big (n^{\frac{d(d-k)}{d-1}}\big )$$ Ω d ( n d ( d - k ) d - 1 ) , which matches the upper bound proved by Balko et al., and settles a problem proposed by Brass et al. Furthermore, we improve the best known lower bound on the maximum number of incidences between points and hyperplanes in $$\mathbb {R}^{d}$$ R d assuming their incidence graph avoids the complete bipartite graph $$K_{c,c}$$ K c , c for some large constant $$c=c(d)$$ c = c ( d ) .

Abstract. In this paper we prove several results on Ramsey numbers [Formula: see text] for a fixed graph [Formula: see text] and a large graph [Formula: see text], in particular for [Formula: see text]. These results extend earlier work of Erdős, Faudree, Rousseau, and Schelp and of Balister, Schelp, and Simonovits on so-called Ramsey size-linear graphs. Among other results, we show that if [Formula: see text] is a subdivision of [Formula: see text] with at least six vertices, then [Formula: see text] for every graph [Formula: see text]. We also conjecture that if [Formula: see text] is a connected graph with [Formula: see text], then [Formula: see text]. The case [Formula: see text] was proved by Erdős, Faudree, Rousseau, and Schelp. We prove the case [Formula: see text].

In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number s > 2, we prove that every graph on n vertices with average degree at least d contains a subgraph of average degree at least s on at most vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. * The full version of the paper can be accessed at https://arxiv.org/abs/2207.02170

For a graph $G$, let $f(G)$ denote the size of the maximum cut in $G$. The problem of estimating $f(G)$ as a function of the number of vertices and edges of $G$ has a long history and was extensively studied in the last fifty years. In this paper we propose an approach, based on semidefinite programming, to prove lower bounds on $f(G)$. We use this approach to find large cuts in graphs with few triangles and in $K_r$-free graphs.

Given a large graph $H$, does the binomial random graph $G(n,p)$ contain a copy of $H$ as an induced subgraph with high probability? This classical question has been studied extensively for various graphs $H$, going back to the study of the independence number of $G(n,p)$ by Erdös and Bollobás and by Matula in 1976. In this paper we prove an asymptotically best possible result for induced matchings by showing that if $C/n\le p \le 0.99$ for some large constant $C$, then $G(n,p)$ contains an induced matching of order approximately $2\log_q(np)$, where $q= \frac{1}{1-p}$.

A celebrated result of Mantel shows that every graph on n vertices with $\lfloor n^2/4 \rfloor + 1$ edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must, in fact, be at least $\lfloor n/2 \rfloor$ triangles in any such graph. Another strengthening, due to the combined efforts of many authors starting with Erdös, says that any such graph must have an edge which is contained in at least $n/6$ triangles. Following Mubayi, we study the interplay between these two results, that is, between the number of triangles in such graphs and their book number, the largest number of triangles sharing an edge. Among other results, Mubayi showed that for any $1/6 \leq \beta < 1/4$ there is $\gamma > 0$ such that any graph on $n$ vertices with at least $\lfloor n^2/4\rfloor + 1$ edges and book number at most $\beta n$ contains at least $(\gamma -o(1))n^3$ triangles. He also asked for a more precise estimate for $\gamma$ in terms of $\beta$. We make a conjecture about this dependency and prove this conjecture for $\beta = 1/6$ and for $0.2495 \leq \beta < 1/4$, thereby answering Mubayi's question in these ranges.

Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it is well known and easy to show that $R(G,H) \geq (|G|-1)(\chi(H)-1)+\sigma(H)$, where $\chi(H)$ is the chromatic number of $H$ and $\sigma(H)$ is the size of the smallest color class in a $\chi(H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)= (|G|-1)(\chi(H)-1)+\sigma(H)$. The notion of Ramsey goodness was introduced by Burr and Erdös in 1983 and has been extensively studied since then. In this paper we show that if $n\geq 10^{60}|H|$ and $\sigma(H)\geq \chi(H)^{22}$, then the $n$-vertex cycle $C_n$ is $H$-good. For graphs $H$ with high $\chi(H)$ and $\sigma(H)$, this proves in a strong form a conjecture of Allen, Brightwell, and Skokan.

A subset $S$ of initially infected vertices of a graph $G$ is called zero forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbor, infects this neighbor. The zero forcing number of $G$ is the minimum cardinality of a zero forcing set in $G$. We study the zero forcing number of various classes of graphs, including graphs of large girth, $H$-free graphs for a fixed bipartite graph $H$, and random and pseudorandom graphs.

Submodular function minimization (SFM) is a fundamental and efficiently solvable problem class in combinatorial optimization with a multitude of applications in various fields. Surprisingly, there is only very little known about constraint types under which SFM remains efficiently solvable. The arguably most relevant non-trivial constraint class for which polynomial SFM algorithms are known are parity constraints, i.e., optimizing only over sets of odd (or even) cardinality. Parity constraints capture classical combinatorial optimization problems like the odd-cut problem, and they are a key tool in a recent technique to efficiently solve integer programs with a constraint matrix whose subdeterminants are bounded by two in absolute value. We show that efficient SFM is possible even for a significantly larger class than parity constraints, by introducing a new approach that combines techniques from Combinatorial Optimization, Combinatorics, and Number Theory. In particular, we can show that efficient SFM is possible over all sets (of any given lattice) of cardinality r mod m, as long as m is a constant prime power. This covers generalizations of the odd-cut problem with open complexity status, and with relevance in the context of integer programming with higher subdeterminants. To obtain our results, we establish a connection between the correctness of a natural algorithm, and the inexistence of set systems with specific combinatorial properties. We introduce a general technique to disprove the existence of such set systems, which allows for obtaining extensions of our results beyond the above-mentioned setting. These extensions settle two open questions raised by Geelen and Kapadia [Combinatorica, 2017] in the context of computing the girth and cogirth of certain types of binary matroids.

A family $\mathcal A$ of subsets of an $n$-element set is called an eventown (resp., oddtown) if all its sets have even (resp., odd) size and all pairwise intersections have even size. Using tools from linear algebra, it was shown by Berlekamp and Graver that the maximum size of an eventown is $2^{\left\lfloor n/2\right\rfloor}$. On the other hand (somewhat surprisingly), it was proven by Berlekamp that oddtowns have size at most $n$. Over the last four decades, many extensions of this even/oddtown problem have been studied. In this paper we present new results on two such extensions. First, extending a result of Vu, we show that a $k$-wise eventown (i.e., intersections of $k$ sets are even) has for $k \geq 3$ a unique extremal configuration and obtain a stability result for this problem. Next we improve some known bounds for the defect version of an $\ell$-oddtown problem. In this problem we consider sets of size $\not\equiv 0 \pmod \ell$ where $\ell$ is a prime number (not necessarily 2) and allow a few pairwise intersections to also have size $\not\equiv 0 \pmod \ell$.

We show that for any fixed dense graph $G$ and bounded-degree tree $T$ on the same number of vertices, a modest random perturbation of $G$ will typically contain a copy of $T$. This combines the viewpoints of the well-studied problems of embedding trees into fixed dense graphs and into random graphs, and extends a sizable body of existing research on randomly perturbed graphs. Specifically, we show that there is $c=c(\alpha,\Delta)$ such that if $G$ is an $n$-vertex graph with minimum degree at least $\alpha n$, and $T$ is an $n$-vertex tree with maximum degree at most $\Delta$, then if we add $cn$ uniformly random edges to $G$, the resulting graph will contain $T$ asymptotically almost surely (as $n\to\infty$). Our proof uses a lemma concerning the decomposition of a dense graph into superregular pairs of comparable sizes, which may be of independent interest.

Let G = (V, E) be a connected undirected graph with k vertices. Suppose that on each vertex of the graph there is a player having an n-bit string. Each player is allowed to communicate with its neighbors according to a (static) agreed communication protocol, and the players must decide, deterministically, if their inputs are all equal. What is the minimum possible total number of bits transmitted in a protocol solving this problem ? We determine this minimum up to a lower order additive term in many cases. In particular, we show that it is kn/2 + o(n) for any Hamiltonian k-vertex graph, and that for any 2-edge connected graph with m edges containing no two adjacent vertices of degree exceeding 2 it is mn/2 + o(n). The proofs combine graph theoretic ideas with tools from additive number theory.

A conjecture of Bandelt and Dress states that the maximum quartet distance between any two phylogenetic trees on n leaves is at most . Using the machinery of flag algebras we improve the currently known bounds regarding this conjecture, in particular we show that the maximum is at most . We also give further evidence that the conjecture is true by proving that the maximum distance between caterpillar trees is at most .

A conjecture of Bandelt and Dress states that the maximum quartet distance between any two phylogenetic trees on $n$ leaves is at most $(\frac{2}{3}+o(1))\binom{n}{4}$. Using the machinery of flag algebras, we improve the currently known bounds regarding this conjecture; in particular, we show that the maximum is at most $(0.69+o(1))\binom{n}{4}$. We also give further evidence that the conjecture is true by proving that the maximum distance between caterpillar trees is at most $(\frac{2}{3}+o(1))\binom{n}{4}$.

Given a random 3-uniform hypergraph $H=H(n,p)$ on $n$ vertices where each triple independently appears with probability $p$, consider the following graph process. We start with the star $G_0$ on the same vertex set, containing all the edges incident to some vertex $v_0$, and repeatedly add an edge $xy$ if there is a vertex $z$ such that $xz$ and $zy$ are already in the graph and $xzy\in H$. We say that the process propagates if it reaches the complete graph before it terminates. In this paper we prove that the threshold probability for propagation is $p=\frac{1}{2\sqrt{n}}$. We conclude that $p=\frac{1}{2\sqrt{n}}$ is an upper bound for the threshold probability that a random 2-dimensional simplicial complex is simply connected.

In the musical chairs game $MC(n,m)$, a team of $n$ players plays against an adversarial scheduler. The scheduler wins if the game proceeds indefinitely, while termination after a finite number of rounds is declared a win of the team. At each round of the game each player occupies one of the $m$ available chairs. Termination (and a win of the team) is declared as soon as each player occupies a unique chair. Two players that simultaneously occupy the same chair are said to be in conflict. In other words, termination (and a win for the team) is reached as soon as there are no conflicts. The only means of communication throughout the game is this: At every round of the game, the scheduler selects an arbitrary nonempty set of players who are currently in conflict, and notifies each of them separately that it must move. A player who is thus notified changes its chair according to its deterministic program. As we show, for $m\ge 2n-1$ chairs the team has a winning strategy. Moreover, using topological arguments we show that this bound is tight. For $m\leq 2n-2$ the scheduler has a strategy that is guaranteed to make the game continue indefinitely and thus win. We also have some results on additional interesting questions. For example, if $m \ge 2n-1$ (so that the team can win), how quickly can they achieve victory?

For natural numbers d and t there exists a positive C such that if F is a family of nC semi-algebraic sets in Rd of description complexity at most t, then there is a subset F' of F of size $n$ such that either every pair of elements in F' intersect or the elements of F' are pairwise disjoint. This result, which also holds if the intersection relation is replaced by any semi-algebraic relation of bounded description complexity, was proved by Alon, Pach, Pinchasi, Radoicic, and Sharir and improves on a bound of 4n for the family F which follows from a straightforward application of Ramsey's theorem. We extend this semi-algebraic version of Ramsey's theorem to k-ary relations and give matching upper and lower bounds for the corresponding Ramsey function, showing that it grows as a tower of height k-1. This improves on a direct application of Ramsey's theorem by one exponential. We apply this result to obtain new estimates for some geometric Ramsey-type problems relating to order types and one-sided sets of hyperplanes. We also study the off-diagonal case, achieving some partial results.

We present an all-pairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from [0,1] is O ( n 2 ), in expectation and with high probability. This resolves a long-standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano [2006]. The analysis relies on a proof that the number of locally shortest paths in such randomly weighted graphs is O ( n 2 ), in expectation and with high probability. We also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in O (log 2 n ) expected time.

An old problem raised independently by Jacobson and Schönheim seeks to determine the maximum $s$ for which every graph with $m$ edges contains a pair of edge-disjoint isomorphic subgraphs with $s$ edges. In this paper we determine this maximum up to a constant factor. We show that every $m$-edge graph contains a pair of edge-disjoint isomorphic subgraphs with at least $c (m\log m)^{2/3}$ edges for some absolute constant $c$, and find graphs where this estimate is off only by a multiplicative constant. Our results improve bounds of Erdös, Pach, and Pyber from 1987.

We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on N vertices with (N2)-o(N2) edges, which can be decomposed into pairwise disjoint induced matchings, each of size N1-o(1). The second construction provides a covering of all edges of the complete graph KN by two graphs, each being the edge disjoint union of at most N2-δ induced matchings, where δ>0.076. This disproves (in a strong form) a conjecture of Meshulam, substantially improves a result of Birk, Linial and Meshulam on communicating over a shared channel, and (slightly) extends the analysis of Hastad and Wigderson of the graph test of Samorodnitsky and Trevisan for linearity. Additionally, our constructions settle a combinatorial question of Vempala regarding a candidate rounding scheme for the directed Steiner tree problem.

Let [Formula: see text] be a sequence of [Formula: see text] integers. For an increasing monotone graph property [Formula: see text] we say that a base graph [Formula: see text] is [Formula: see text]-resilient with respect to [Formula: see text] if for every subgraph [Formula: see text] such that [Formula: see text] for every [Formula: see text] the graph [Formula: see text] possesses [Formula: see text]. This notion naturally extends the idea of the local resilience of graphs recently initiated by Sudakov and Vu. In this paper we study the [Formula: see text]-resilience of a typical graph from [Formula: see text] with respect to the Hamiltonicity property, where we let [Formula: see text] range over all values for which the base graph is expected to be Hamiltonian. Considering this generalized approach to the notion of resilience our main result implies several corollaries which improve on the best known bounds of Hamiltonicity related questions. For one, it implies that for every positive [Formula: see text] and large enough values of [Formula: see text], if [Formula: see text], then with high probability the local resilience of [Formula: see text] with respect to being Hamiltonian is at least [Formula: see text], improving on the previous bound for this range of [Formula: see text]. Another implication is a result on optimal packing of edge-disjoint Hamilton cycles in a random graph. We prove that if [Formula: see text], then with high probability a graph [Formula: see text] sampled from [Formula: see text] contains [Formula: see text] edge-disjoint Hamilton cycles, extending the previous range of [Formula: see text] for which this was known to hold.

In this short paper we study the game of cops and robbers, which is played on the vertices of some fixed graph [Formula: see text]. Cops and a robber are allowed to move along the edges of [Formula: see text], and the goal of cops is to capture the robber. The cop number [Formula: see text] of [Formula: see text] is the minimum number of cops required to win the game. Meyniel conjectured a long time ago that [Formula: see text] cops are enough for any connected [Formula: see text] on [Formula: see text] vertices. Improving several previous results, we prove that the cop number of an [Formula: see text]-vertex graph is at most [Formula: see text]. A similar result independently and slightly before us was also obtained by Lu and Peng.

We present an all-pairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from [0,1] is O(n2), in expectation and with high probability. This resolves a long standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano. The analysis relies on a proof that the number of locally shortest paths in such randomly weighted graphs is O(n2), in expectation and with high probability. We also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in O(log2n) expected time.

We present new explicit constructions of deterministic randomness extractors, dispersers and related objects. We say that a distribution X on binary strings of length n is a δ-source if X assigns probability at most 2 −δ n to any string of length n . For every δ>0, we construct the following poly( n )-time computable functions: 2-source disperser: D:({0, 1} n ) 2 → {0, 1} such that for any two independent δ-sources X 1 , X 2 we have that the support of D ( X 1 , X 2 ) is {0, 1}. Bipartite Ramsey graph: Let N =2 n . A corollary is that the function D is a 2-coloring of the edges of K N,N (the complete bipartite graph over two sets of N vertices) such that any induced subgraph of size N δ by N δ is not monochromatic. 3-source extractor: E :({0, 1} n ) 3 → {0, 1} such that for any three independent δ-sources X 1 , X 2 , X 3 we have that E ( X 1 , X 2 , X 3 ) is o (1)-close to being an unbiased random bit. No previous explicit construction was known for either of these for any δ<1/2, and these results constitute significant progress to long-standing open problems. A component in these results is a new construction of condensers that may be of independent interest: This is a function C :{0, 1} n → ({0, 1} n/c ) d (where c and d are constants that depend only on δ) such that for every δ-source X one of the output blocks of C(X) is (exponentially close to) a 0.9-source. (This result was obtained independently by Ran Raz.) The constructions are quite involved and use as building blocks other new and known objects. A recurring theme in these constructions is that objects that were designed to work with independent inputs, sometimes perform well enough with correlated, high entropy inputs. The construction of the disperser is based on a new technique which we call “the challenge-response mechanism” that (in some sense) allows “identifying high entropy regions” in a given pair of sources using only one sample from the two sources.

A graph G on n vertices is pancyclic if it contains cycles of length t for all $3\leq t\leq n$. In this paper we prove that for any fixed $\epsilon>0$, the random graph $G(n,p)$ with $p(n)\gg n^{-1/2}$ (i.e., with $p(n)/n^{-1/2}$ tending to infinity) asymptotically almost surely has the following resilience property. If H is a subgraph of G with maximum degree at most $(1/2-\epsilon)np$, then $G-H$ is pancyclic. In fact, we prove a more general result which says that if $p\gg n^{-1+1/(l-1)}$ for some integer $l\geq3$, then for any $\epsilon>0$, asymptotically almost surely every subgraph of $G(n,p)$ with minimum degree greater than $(1/2+\epsilon)np$ contains cycles of length t for all $l\leq t\leq n$. These results are tight in two ways. First, the condition on p essentially cannot be relaxed. Second, it is impossible to improve the constant $1/2$ in the assumption for the minimum degree. We also prove corresponding results for pseudorandom graphs.

For a real $c \geq 1$ and an integer n, let $f(n,c)$ denote the maximum integer f such that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in particular, every graph on n vertices contains a regular induced subgraph on at least $f(n,1)$ vertices. The problem of estimating $f(n,1)$ was posed long ago by Erdős, Fajtlowicz, and Staton. In this paper we obtain the following upper and lower bounds for the asymptotic behavior of $f(n,c)$: (i) For fixed $c>2.1$, $n^{1-O(1/c)} \leq f(n,c) \leq O(cn/\log n)$. (ii) For fixed $c=1+\varepsilon$ with $\varepsilon>0$ sufficiently small, $f(n,c) \geq n^{\Omega(\varepsilon^2/ \ln (1/\varepsilon))}$. (iii) $\Omega (\ln n) \leq f(n,1) \leq O(n^{1/2} \ln^{3/4} n)$. An analogous problem for not necessarily induced subgraphs is briefly considered as well.

In this short note we prove that there is a constant c such that every k-edge-coloring of the complete graph $K_n$ with $n \geq 2^{ck}$ contains a $K_4$ whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and is the first exponential bound for this problem.

For two graphs H and G, the Ramsey number $r(H, G)$ is the smallest positive integer n such that every red-blue edge coloring of the complete graph $K_n$ on n vertices contains either a red copy of H or a blue copy of G. Motivated by questions posed by Erdős and Harary, in this note we study how the Ramsey number $r(K_s, G)$ depends on the size of the graph G. For $s \geq 3$, we prove that for every G with m edges, $r(K_s,G) \geq c(m/\log m)^{(s+1)/(s+3)}$ for some positive constant c depending only on s. This lower bound improves an earlier result of Erdős, Faudree, Rousseau, and Schelp, and it is tight up to a polylogarithmic factor when $s=3$. We also study the maximum value of $r(K_s,G)$ as a function of m.

A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E/sub P/'(G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. 1) For any /spl epsiv/ > 0 and any monotone property P, there is a deterministic algorithm, which given a graph G of size n, approximates E/sub P/'(G) in time O(n/sup 2/) to within an additive error of /spl epsiv/n/sup 2/. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E/sub P/'. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist; 1. If there is a bipartite graph that does not satisfy P, then there is a /spl delta/ > 0 for which it is possible to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta// in polynomial time. 2) On the other hand, if all bipartite graphs satisfy P, then for any /spl delta/ > 0 it is NP-hard to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta//. While the proof of (1) is simple, the proof of (2) requires several new ideas and involves tools from extremal graph theory together with spectral techniques. This approach may be useful for obtaining other hardness of approximation results. Interestingly, prior to this work it was not even known that computing E/sub P/' precisely for the properties in (2) is NP-hard. We thus answer (in a strong form) a question of Yannakakis [1981], who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing E/sub P/' is NP-hard.

A distribution X over binary strings of length n has min-entropy k if every string has probability at most 2-k in X. We say that X is a δ-source if its rate k⁄n is at least δ.We give the following new explicit instructions (namely, poly(n)- time computable functions) of deterministicextractors, 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.

The strong chromatic index of a graph G, denoted by $\chi_s(G)$, is the minimum number of colors needed to color its edges so that each color class is an induced matching. In this paper we analyze the asymptotic behavior of this parameter in a random graph $G(n,p)$, for two regions of the edge probability $p=p(n)$. For the dense case, where p is a constant, $0 < p < 1$, we prove that with high probability $\chi_s(G)\le (1+o(1))\frac{3}{4}\frac{n^2p}{\log_bn}$, where $b=1/(1-p)$. This improves upon a result of Czygrinow and Nagle [{\it Discrete Math.}, 281 (2004), pp. 129--136]. For the sparse case, where $np< \frac{1}{100}\sqrt{\log n/\log\log n}$, we show that with high probability $\chi_s(G)=\Delta_1(G)$, where $\Delta_1(G)=\max\{d(u)+d(v)-1:\ (u,v)\in E(G)\}$. This improves a result of Palka [{\it Australas. J. Combin.}, 18 (1998), pp. 219--226].

A set system is L-intersecting if any pairwise intersection size lies in L, where L is some set of s nonnegative integers. The celebrated Frankl--Ray-Chaudhuri--Wilson theorems give tight bounds on the size of an L-intersecting set system on a ground set of size n. Such a system contains at most $\binom{n}{s}$ sets if it is uniform and at most $\sum_{i=0}^s \binom{n}{i}$ sets if it is nonuniform. They also prove modular versions of these results. We consider the following extension of these problems. Call the set systems $\mathcal{A}_1,\ldots,\mathcal{A}_k$ {\em L-cross-intersecting} if for every pair of distinct sets A,B with $A \in \mathcal{A}_i$ and $B \in \mathcal{A}_j$ for some $i \neq j$ the intersection size $|A \cap B|$ lies in L. For any k and for n > n 0 (s) we give tight bounds on the maximum of $\sum_{i=1}^k |\mathcal{A}_i|$. It is at most $\max\, \{k\binom{n}{s}, \binom{n}{\lfloor n/2 \rfloor}\}$ if the systems are uniform and at most $ \max\, \{k sum_{i=0}^s \binom{n}{i} , (k-1) \sum_{i=0}^{s-1} \binom{n}{i} + 2^n\}$ if they are nonuniform. We also obtain modular versions of these results. Our proofs use tools from linear algebra together with some combinatorial ideas. A key ingredient is a tight lower bound for the rank of the inclusion matrix of a set system. The s*-inclusion matrix of a set system $\mathcal{A}$ on [n] is a matrix M with rowsindexed by $\mathcal{A}$ and columns by the subsets of [n] of size at most s, where if $A \in \mathcal{A}$ and $B \subset [n]$ with $|B| \leq s$, we define M AB to be 1 if $B \subset A$ and 0 otherwise. Our bound generalizes the well-known result that if $|\mathcal{A}| < 2^{s+1}$, then M has full rank $|\mathcal{A}|$. In a combinatorial setting this fact was proved by Frankl and Pach in the study of null t-designs; it can also be viewed as determining the minimum distance of the Reed--Muller codes.

We consider the problem of learning a matching (i.e., a graph in which all vertices have degree 0 or 1) in a model where the only allowed operation is to query whether a set of vertices induces an edge. This is motivated by a problem that arises in molecular biology. In the deterministic nonadaptive setting, we prove a $(\frac{1}{2}+o(1)){n \choose 2} $ upper bound and a nearly matching $0.32{n \choose 2}$ lower bound for the minimum possible number of queries. In contrast, if we allow randomness, then we obtain (by a randomized, nonadaptive algorithm) a much lower O(n log n) upper bound, which is best possible (even for randomized fully adaptive algorithms).

We prove a general recursive inequality concerning /spl mu//sup */(R), the asymptotic (least) density of the best binary covering codes of radius R. In particular, this inequality implies that /spl mu//sup */(R)/spl les/e/spl middot/(RlogR+logR+loglogR+2), which significantly improves the best known density 2/sup R/R/sup R/(R+1)/R!. Our inequality also holds for covering codes over arbitrary alphabets.

We consider the problem of learning a matching (i.e., a graph in which all vertices have degree 0 or 1) in a model where the only allowed operation is to query whether a set of vertices induces an edge. This is motivated by a problem that arises in molecular biology. In the deterministic nonadaptive setting, we prove a ( 1/2 +o(1))(n/2) upper bound and a nearly matching 0.32(n/2) lower bound for the minimum possible number of queries. In contrast, if we allow randomness then we obtain (by a randomized, nonadaptive algorithm) a much lower O(n log n) upper bound, which is best possible (even for randomized fully adaptive algorithms).

Semidefinite programming based approximation algorithms, such as the Goemans and Williamson approximation algorithm for the MAX CUT problem, are usually shown to have certain performance guarantees using local ratio techniques. Are the bounds obtained in this way tight? This problem was considered before by Karloff [SIAM J. Comput., 29 (1999), pp. 336--350] and by Alon and Sudakov [ Combin. Probab. Comput., 9 (2000), pp. 1--12]. Here we further extend their results and show, for the first time, that the local analyses of the Goemans and Williamson MAX CUT algorithm, as well as its extension by Zwick, are tight for every possible relative size of the maximum cut in the sense that the expected value of the solutions obtained by the algorithms may be as small as the analyses ensure. We also obtain similar results for a related problem. Our approach is quite general and could possibly be applied to some additional problems and algorithms.