Result: On shifted products which are powers

The author proves: Let \(V=\{x^k: x \in \mathbb N\), \(k \geq 2\}\) be the set of perfect powers. Let \({\mathcal A}\subset \{1,\ldots , N\}\) be a set with the property that \(a_i a_j +1 \in V\), for all pairs \(a_i, a_j \in {\mathcal A},\;a_i \neq a_j\). Then \(|{\mathcal A}| \ll (\frac{\log N}{\log \log N})^{\frac{3}{2}}\). This improves upon the work of \textit{K. Gyarmati, A. Sárközy} and \textit{C. L. Stewart} [Mathematika 49, No. 1--2, 227--230 (2002; Zbl 1047.11029)] and \textit{Y. Bugeaud} and \textit{K. Gyarmati} [Ill. J. Math. 48, No. 4, 1105--1115 (2004; Zbl 1065.11015)] who proved an upper bound of \(O((\frac{\log N}{\log \log N})^2)\). Assuming the ABC conjecture it is proved that \(| {\mathcal A}| =O(1)\) holds, improving a result of \textit{R. Dietmann, C. Elsholtz, K. Gyarmati} and \textit{M. Simonovits} [J. Comb. Theory, Ser. A 111, No. 1, 24--36 (2005; Zbl 1131.11013)]. As in the previous work, the proof makes use of graph theoretic arguments from Ramsey theory. The new ingredient is a more efficient treatment of the large exponents, using bounds for linear forms in logarithms of algebraic numbers.