Abstract
Nous présentons une nouvelle démonstration courte du théorème de van der Waerden polynomial canonique, récemment établi par Girão.
Highlights
Our proof of Theorem 1 follows the strategy of Erdos and Graham [2], who deduced a canonical van der Waerden theorem using Szemerédi’s theorem [7]
Let n be an integer on the order of N 1/D so that x + p1(y), . . . , x + pk (y) ∈ [N ] only if y ∈ [n]
Summing over all i = j, we see that the number of pairs (x, y) ∈ N × [n] where at least two of x + p1(y), . . . , x + pk (y) lie in A is O(ε1/8D−1 |A|n)
Summary
Girão [4] recently proved the following canonical version of the polynomial van der Waerden theorem. Canonical [3] refers to the fact that the statement is independent of the number of colors. A set is rainbow if all elements have distinct colors.
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