Extremal Chowla sets and their linear analogues: A human-AI mathematical investigation using Co-Scientist

We introduce an extremal invariant associated with Chowla-type order conditions in finite groups. A nonempty subset S of a finite group G is called a Chowla set if every element of S has order greater than |S|, and we write \Ccal(G) for the maximum cardinality of such a set. We first show that \Ccal(G) is determined by the distribution of element orders in G. For cyclic groups, we derive an exact divisor formula and characterize the integers n for which \Ccal(\mathbb Z/n\mathbb Z)=φ(n). We prove that \liminf_{n\to\infty}\Ccal(\mathbb Z/n\mathbb Z)/φ(n)=1, whereas \limsup_{n\to\infty}\Ccal(\mathbb Z/n\mathbb Z)/φ(n)=\infty, and we determine the corresponding lower and upper limits under normalization by n. For finite abelian groups, we obtain an explicit formula in terms of the invariant-factor decomposition, together with a closed formula for finite abelian p-groups. We then develop a linear analogue for finite field extensions. A nonzero K-subspace A of an extension L/K is called a Chowla subspace if [K(a):K]>\dim_K A for every nonzero a\in A. Since this condition depends on \dim_K A, it does not generally require every nonzero element of A to generate L over K. Nevertheless, when L/K is finite and separable, we prove the exact formula \Ccal(L/K)=[L:K]-d_{\max}(L/K), where d_{\max}(L/K) is the largest degree over K of a proper intermediate field. For finite fields, we give a direct proof in every degree using a normal-basis construction. This work was developed through an expert-guided human--AI collaboration. A reasoning-focused configuration of Co-Scientist was used to explore examples and potential proof strategies. The authors formulated the problem, independently verified and completed all arguments, and wrote the final proofs.