By Grotschel M , Katona G O (eds )

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4. Remarks. 3). Theorem 1 states, very roughly speaking, that the Surplus is “around” the Core-Density. Since the Core-Density is exactly the maximum local density of a graph, Theorem 1 clariﬁes the vague statement that “dense graphs have large surplus”. It is worth mentioning that the Core-Density (and the Core-Degree) is almost identical to some other well-known graph-theoretic concepts such as (1) the Arboricity, and (2) the Greedy Coloring Number, or Degeneracy. The Arboricity of graph G (here G can be any multigraph) is the forestpartition number, that is, the minimum number of forests (set of disjoint trees) forming a partition of G.

Then let n U denote the sum of all uij with ij ≤ n. This unusual notation will be very convenient. 1 is that U can be partitioned into two subsequences U + and U − such that for every n ∈ N U − ∈ 2dB, U+ − n U− − and also n n U + ∈ 2dB. n The two statements here are equivalent since the norm is symmetric. Further, of course, U + (U − ) consists of elements of U for which ε = +1 (ε = −1). Adding nU to both sides and dividing by 2 gives U + ∈ dB + n 1 2 U U − ∈ dB + and n n 1 2 U. 1 is this.

446, and so the largest clique size that Maker can build (Forcer can force Avoider to build) is 19, 999, 999, 933. This level of accuracy is particularly striking, because for smaller values of n I don’t know the Clique Achievement Number. ); if n = 2100 62 J. Beck then it can be either 99 or 100 or 101 or . . ), that is, there are 90 possible candidates. Even less is known about the small Avoidance Numbers. ) never know the exact values of these game numbers for n = 20, or for n = 100, or for n = 2100 , 10 but we know the exact value for a monster number like n = 210 .