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Combinatorica, 2020
Let p ∈ ℤ [x] be any polynomial with p(0) =0, k ∈ ℕ and let c1, …, cs ∈ ℤ, s ⩾ k(k + 1), be non-zero integers such that $$\sum {{c_1} = 0} $$ . We show that for a wide class of coefficients c1, …, cs in every finite coloring
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Let p ∈ ℤ [x] be any polynomial with p(0) =0, k ∈ ℕ and let c1, …, cs ∈ ℤ, s ⩾ k(k + 1), be non-zero integers such that $$\sum {{c_1} = 0} $$ . We show that for a wide class of coefficients c1, …, cs in every finite coloring
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Random Structures & Algorithms, 1992
AbstractSay a graph H selects a graph G if given any coloring of H, there will be a monochromatic induced copy of G in H or a completely multicolored copy of G in H. Denote by s(G) the minimum order of a graph that selects G and set s(n) = max {s(G): |G| = n}. Upper and lower bounds are given for this function. Also, consider the Folkman function fr(n)
Eaton, Nancy, Rödl, Vojtěch
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AbstractSay a graph H selects a graph G if given any coloring of H, there will be a monochromatic induced copy of G in H or a completely multicolored copy of G in H. Denote by s(G) the minimum order of a graph that selects G and set s(n) = max {s(G): |G| = n}. Upper and lower bounds are given for this function. Also, consider the Folkman function fr(n)
Eaton, Nancy, Rödl, Vojtěch
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Graphs and Combinatorics, 1985
An edge-coloring of a graph is a partition of the set of edges into color-classes such that no two edges in the same class are adjacent. A subsetA of the vertex set isantihomogeneous if all edges in the subgraph induced byA have different colors.
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An edge-coloring of a graph is a partition of the set of edges into color-classes such that no two edges in the same class are adjacent. A subsetA of the vertex set isantihomogeneous if all edges in the subgraph induced byA have different colors.
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The independence of Ramsey's theorem
Journal of Symbolic Logic, 1969In [3] F. P. Ramsey proved as a theorem of Zermelo-Fraenkel set theory (ZF) with the Axiom of Choice (AC) the following result:(1) Theorem. Let A be an infinite class. For each integer n and partition {X, Y} of the size n subsets of A, there exists an infinite subclass of A ...
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Graphs and Combinatorics, 1985
Letn?k?1 be integers and letf(n, k) be the smallest integer for which the following holds: If ? is a family of subsets of ann-setX with |?|
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Letn?k?1 be integers and letf(n, k) be the smallest integer for which the following holds: If ? is a family of subsets of ann-setX with |?|
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2013
The basic problem of (combinatorial) discrepancy theory is how to color a set with two colors as uniformly as possible with respect to a given family of subsets. The aim is to achieve that each of the two colors meets each subset under consideration in approximately the same number of elements. From the finite Ramsey theorem (cf. Corollary 7.2) we know
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The basic problem of (combinatorial) discrepancy theory is how to color a set with two colors as uniformly as possible with respect to a given family of subsets. The aim is to achieve that each of the two colors meets each subset under consideration in approximately the same number of elements. From the finite Ramsey theorem (cf. Corollary 7.2) we know
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