Ascending waves

On three sets with nondecreasing diameter

“…Subsequently, Bollobás, Erdős, and Jin obtained improved results for m = 2, showing that 4r − log 2 r + 1 ≤ f * (2, 2; r) ≤ 4r + 1 and f * (2, 2; 2 k ) = 4 · 2 k + 1, as well as giving improved asymptotic bounds for t and r when m = 2. The value of f (m, m; 4) was determined to be 12m − 9 in [8], the off diagonal cases (when not all m i = m) are introduced in [14], and other related Ramsey-type problems can also be found in [1] [5] [10] [11] [12] [13]. The goal of this paper is to improve the estimates from (1) to the first exact value for more than two sets.…”

“…The value of f (m, m; 4) was determined to be 12m − 9 in [8], the off diagonal cases (when not all m i = m) are introduced in [14], and other related Ramsey-type problems can also be found in [1] [5] [10] [11] [12] [13]. The goal of this paper is to improve the estimates from (1) to the first exact value for more than two sets. Indeed, we will show that both the upper and lower bounds of Bialostocki Let ∆ : X → C be a coloring of a finite set X by a set of colors C. Let c ∈ C and Y ⊆ X.…”

On monochromatic ascending waves

“…, w k/2 we have w k/2 − w k/2−1 ≥ O(k 2r−2−ǫ/2 ). The following definition and lemma, which are generalizations of those found in [1], will give us the desired result.…”

“…We believe that AW (k; r) = O(k 2r−1 ), however, we have thus far been unable to prove this. The approach of Alon and Spencer [1], which is to show that there exists an r-coloring (under a random coloring scheme) such that every monochromatic 3k 4 -term ascending wave has d 3k/4−1 > ck 2r−2 does not work for an arbitrary number of colors with our generalization. However, for 3 colors, we can refine their argument to prove that AW (k; 3) = O(k 5 ).…”

“…Additionally, we show that for fixed k ≥ 3, (1)). Soon after, Alon and Spencer [1] showed that for sufficiently large k, AW (k; 2) > k 3 10 21 − k 2 10 20 − k 10 + 4.…”