Exponential bounds for the Erdős-Ginzburg-Ziv constant

Erdős-Ginzburg-Ziv Constants by Avoiding Three-Term Arithmetic Progressions

“…, which slightly improves the previously best known bounds for g(F n p ) and s(F n p ) from [19]. To obtain an upper bound for g(F n p ) in terms of r(F n p ), note that a product construction shows…”

“…The case G = F n p for a prime p ≥ 3 has attracted particular interest. In this case, Naslund [19] proved that g(F n p ) ≤ (2 p − p − 2) · (J(p)p) n and s(F n p ) ≤ (p − 1)2 p · (J(p)p) n , where 0.8414 ≤ J(p) ≤ 0.9184. To prove these bounds, Naslund introduced a variant of Tao's slice rank method [22].…”

A lower bound for the k‐multicolored sum‐free problem in Zmn

“…Thus, L k · (Γ m,k ) n , and we can conclude L (Γ m,k ) n by taking → ∞. Lemma 9.2 has a standard proof, it was given for example in [8,Proposition 4.12], see also [26,Lemma 5]. For the reader's convenience, we repeat the proof here:…”

“…Furthermore, the function tends to infinity when γ → 0, hence the minimum value Γ m,k is indeed attained for some 0 < γ m,k < 1 (one can show that there is a unique 0 < γ m,k < 1 where the minimum is attained, but this is not necessary for our purposes). Tao's slice rank method [35], together with the arguments from Blasiak et al [8], gives the following upper bound for the size of k-colored sum-free sets in Z n m for prime powers m. For the reader's convenience, we give a proof of Theorem 1.2 in Section 9 (see also [26,Theorem 4], which is a very similar theorem). Theorem 1.2.…”

Monochromatic equilateral triangles in the unit distance graph

“…The slice rank method has seen a wide array of applications [2, 3, 9 13–15, 18], and we refer the reader to [2, Section 4] for an in‐depth discussion of the properties of the slice rank. For our purposes, we will need the critical lemma, which was proven by Tao.…”