We address the problem of finding sets of integers of a given size with maximum number of pairs summing to powers of $2$. By fixing particular pairs this problem reduces to finding a labeling of the vertices of a given graph with pairwise distinct integers such that the endpoint labels for each edge sum to a power of $2$. We propose an efficient algorithm for this problem, which we use to determine the maximum size of graphs of order $n$ that admit such a labeling for all $n\leq 18$. We also identify the minimal forbidden subgraphs of order $\leq 11$, whose presence prevents graphs from having such a labeling.

翻译：我们研究在给定大小的整数集合中，寻找使得和为2的幂的数对数量最大的问题。通过固定特定数对，该问题可归约为：对给定图的顶点赋予两两不同的整数标签，使得每条边两端标签之和为2的幂。我们针对该问题提出了一种高效算法，并利用它确定了所有阶数n≤18的图中，能够实现此类标签的最大图规模。同时，我们还识别出阶数≤11的最小禁用子图——这些子图的存在将阻止原图具备此类标签。