The passage discusses two theorems related to the distribution of squarefree integers and the existence of equilibrium points in social systems. The first theorem states that if \( y(x) \) is a function such that \( \lim_{x \to \infty} y(x) = \infty \), then the number of integers \( n \leq z \) for which the interval \( (n, n+y(n)) \) contains no squarefree integer is \( o(z) \) as \( z \to \infty \). The second theorem asserts that if \( \lim_{x \to \infty} y(x) = \infty \) and \( y(x) / y(\lambda x) \) is bounded for some \( 0 < \lambda < 1 \), then the normal order of \( \sum_{x \leq n \leq x+y} \mu^2(n) \) is \( (6/\pi^2)y \). The passage also compares these results with those for all integers, noting that the formula \( \sum_{n \leq x} \mu^2(n) = \frac{6x}{\pi^2} + O(\sqrt{x}) \) provides an interval \( [x, x+k\sqrt{x}] \) that always contains a squarefree number for sufficiently large \( k \). This result has been refined to \( [x, x+kx^{1/4}] \) and even \( [x, x+kx^{1/4}/\log x] \) by various mathematicians, with Roth improving it to \( [x, x+x^{1/n+1}] \). Additionally, the passage introduces a social equilibrium existence theorem by Gerard Debreu, which provides conditions under which a social system has an equilibrium. The theorem is used to prove the existence of equilibrium in classical competitive economic systems and in multi-person games. The proof involves topological concepts and a fixed point theorem, showing that if certain conditions are met, an equilibrium point exists. The theorem is applied to saddle points and the MinMax operator, and a historical note is provided, tracing the development of the concept of equilibrium points in economics and game theory.The passage discusses two theorems related to the distribution of squarefree integers and the existence of equilibrium points in social systems. The first theorem states that if \( y(x) \) is a function such that \( \lim_{x \to \infty} y(x) = \infty \), then the number of integers \( n \leq z \) for which the interval \( (n, n+y(n)) \) contains no squarefree integer is \( o(z) \) as \( z \to \infty \). The second theorem asserts that if \( \lim_{x \to \infty} y(x) = \infty \) and \( y(x) / y(\lambda x) \) is bounded for some \( 0 < \lambda < 1 \), then the normal order of \( \sum_{x \leq n \leq x+y} \mu^2(n) \) is \( (6/\pi^2)y \). The passage also compares these results with those for all integers, noting that the formula \( \sum_{n \leq x} \mu^2(n) = \frac{6x}{\pi^2} + O(\sqrt{x}) \) provides an interval \( [x, x+k\sqrt{x}] \) that always contains a squarefree number for sufficiently large \( k \). This result has been refined to \( [x, x+kx^{1/4}] \) and even \( [x, x+kx^{1/4}/\log x] \) by various mathematicians, with Roth improving it to \( [x, x+x^{1/n+1}] \). Additionally, the passage introduces a social equilibrium existence theorem by Gerard Debreu, which provides conditions under which a social system has an equilibrium. The theorem is used to prove the existence of equilibrium in classical competitive economic systems and in multi-person games. The proof involves topological concepts and a fixed point theorem, showing that if certain conditions are met, an equilibrium point exists. The theorem is applied to saddle points and the MinMax operator, and a historical note is provided, tracing the development of the concept of equilibrium points in economics and game theory.