The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, algebraic and probabilistic methods, and discuss several intriguing conjectures and open problems. This is mainly a survey of recent and less recent results in the area, but it contains several new results as well.

We have sorted the problems into sections: A. Taking and Breaking B. Pushing and Placing Pieces C. Playing with Pencil and Paper D. Disturbing and Destroying ... They have been given new numbers. The numbers in parentheses are the old numbers used in each of the lists of unsolved problems given on pp. 183–189 of AMS Proc. Sympos. Appl. Math. 43 (1991), called PSAM 43 below; on pp. 475–491 of Games of No Chance, hereafter referred to as GONC; and on pp. 457–473 of More Games of No Chance (MGONC). Missing numbers are of problems which have been solved, or for which we have nothing new to add. References [year] may be found in Fraenkel’s Bibliography at the end of this volume. References [#] are at the end of this article. A useful reference for the rules and an introduction to many of the specific games mentioned below is

the box argument or the chest of drawers argument (Schubfachprinzip) can be described, rather vaguely, as follows. If sufficiently many objects are distributed over not too many classes, then at least one class contains many of these objects. In 1930 F. P. Ramsey [12] discovered

The Set Covering problem (SCP) is a well known combinatorial optimization problem, which is NP-hard. We conducted a comparative study of eight different approximation algorithms for the SCP, including several greedy variants, fractional relaxations, randomized algorithms and a neural network algorithm. The algorithms were tested on a set of random-generated problems with up to 500 rows and 5000 columns, and on two sets of problems originating in combinatorial questions with up to 28160 rows and 11264 columns. On the random problems and on one set of combinatorial problems, the best algorithm among those we tested was the neural network algorithm, with greedy variants very close in second and third place. On the other set of combinatorial problems, the best algorithm was a greedy variant and the neural network performed quite poorly. The other algorithms we tested were always inferior to the ones mentioned above. Theoretical Division and CNLS, MS B-213 Los Alamos National Lab, Los Ala...

We introduce the notion of covering complexity of a verifier for probabilistically checkable proofs (PCP). Such a verifier is given an input, a claimed theorem, and an oracle, representing a purported proof of the theorem. The verifier is also given a random string and decides whether to accept the proof or not, based on the given random string. We define the covering complexity of such a verifier, on a given input, to be the minimum number of proofs needed to “satisfy ” the verifier on every random string, i.e., on every random string, at least one of the given proofs must be accepted by the verifier. The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems, and in particular, (hyper-)graph coloring problems. We present a PCP verifier for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a super-constant covering complexity for statements not in the language. Moreover, the acceptance predicate of this verifier is a simple Not-all-Equal check on the four bits it reads. This enables us to prove that for any constant c, it is NP-hard to color a 2-colorable 4-uniform hypergraph using just c colors, and also yields a super-constant inapproximability result under a stronger hardness

Abstract. A permutation is called layered if it consists of the disjoint union of substrings (layers) so that the entries decrease within each layer, and increase between the layers. We find the generating function for the number of permutations on n letters avoiding (1, 2, 3) and a layered permutation on k letters. In the most interesting case of two layers, the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.

Abstract. Many NP-complete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2-colorability (Property B) and random Not-All-Equal k-SAT is 2 k−1 ln 2 − O(1). As a corollary, we establish that the threshold for random k-SAT is of order Θ(2 k), resolving a long-standing open problem.

Let g(n) denote the least integer such that among any g(n) points in general position in the plane there are always n in convex position. In 1935 P. Erdos and G. Szekeres showed that g(n) exists and 2 n\Gamma2 + 1 g(n) i 2n\Gamma4 n\Gamma2 j + 1. Recently, the upper bound has been slightly improved by Chung and Graham and by Kleitman and Pachter. In this note we further improve the upper bound to g(n) / 2n \Gamma 5 n \Gamma 2 ! + 2: 1 Introduction In the 1930's Esther Klein raised the following question. Is it true that for every n there is a least number g(n) such that among any g(n) points in general position in the plane there are always n in convex position? Her question was answered in the affirmative in a classical paper of Erdos and Szekeres [ES35]. In fact, they showed [ES35, ES60] that 2 n\Gamma2 + 1 g(n) / 2n \Gamma 4 n \Gamma 2 ! + 1: The lower bound, 2 n\Gamma2 + 1, is sharp for n = 2; 3; 4; 5 and has been conjectured to be sharp for all n. Ho...

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We prove that, for every family F of n semi-algebraic sets in R d of constant description complexity, there exist a positive constant ε that depends on the maximum complexity of the elements of F, and two subfamilies F1, F2 ⊆ F with at least εn elements each, such that either every element of F1 intersects all elements of F2 or no element of F1 intersects any element of F2. This implies the existence of another constant δ such that F has a subset F ′ ⊆ F with n δ elements, so that either every pair of elements of F ′ intersect each other or the elements of F ′ are pairwise disjoint. The same results hold when the intersection relation is replaced by any other semi-algebraic relation. We apply these results to settle several problems in discrete geometry and in Ramsey theory. 1