A family of concepts involving maximum independent sets which have been under recent study are introduced. Interrelations among the concepts are pointed out and some recent relevant work is discussed.
A family of concepts involving maximum independent sets which have been under recent study are introduced. Interrelations among the concepts are pointed out and some recent relevant work is discussed.
Let G be a transitive permutation group on a set Ω of v points {1, 2, …, v}. Let H be an intransitive subgroup of G and let Δ a set of k points where Δ consists of complete orbits of H. Then the images Δx of Δ under permutations x of Δ have been shown by the first author to be a partially balanced block design D with G as a group of automorphisms. Under certain circumstances D is a balanced incomplete block design. Here a representation of the simple group PSL3(4) of order 20,160 on 56 letters leads to a new symmetric block design with parameters v=56, k-11, λ=2. A representation of the simple group of order 25,920 as U4(4) on 45 isotropic points gives a symmetric design with v=45, k=12, λ=3. One representation of U4(4) on 40 points, gives the design of planes in PG(3, 3) and exhibits the isomorphism of this group to the symplectic group S4(3).
The van der Waerden permanent conjecture is shown to belong to a large family of conjectured inequalities which are of some interest in themselves and all of which might be provable by a routine computation with convex bodies. In fact, the permanent conjecture in cases n=3 and 4 does yield to this method. For n=5, with the computations made by hand, no proof was found, but a slight extension of the computation (which would probably require electronic assistance) could still settle this case and perhaps even a few more small values of n. The question of whether the method must work remains open.
A “random” n-person non-cooperative game—the game that prohibits communication and therefore coalitions among the n players—is shown to have with high probability a pure strategy solution. Such a solution is by definition an equilibrium point or a set of strategies, one for each player, such that if n−1 players use their equilibrium strategies then the n-th player has no reason to deviate from his equilibrium strategy. It is shown that the probability of a solution in pure strategies for large random n-person games converges to (1−1/e) for all n≥2.
This paper gives a proof of the fact that the chromatic number of an orientable surface of genus p is equal to the integral part of whenever the latter is congruent to 2 modulo 12.
In this note a multinomial extension of a result of Knuth is presented which allows very simple proofs of complicated multinomial coefficient identities. Proofs are given of the “multinomial theorem” and of Hurwitz's generalization of Abel's identity, the latter a well-known generlization of the binomial theorem.
This paper gives a proof of the fact that the chromatic number of an orientable surface of genus p is equal to the integral part of whenever the latter s congruent to 8 modulo 12.
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