GPT 6 Astra Ultra attempt on Math-5880
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Contents
Erdős Problem #765: the asymptotic $C_4$ extremal number
Statement and outcome
Determine the asymptotic value of $\operatorname{ex}(n,C_4)$, the maximum edge count in a simple $n$-vertex graph containing no four-cycle. We reconstruct the classical answer \[ \operatorname{ex}(n,C_4)=\left(\frac12+o(1)\right)n^{3/2}. \] Attempt status: solved. The lower-order refinements mentioned in the editorial are separate questions.Counting common neighbours
In a $C_4$-free graph, two distinct vertices have at most one common neighbour. Counting unordered pairs of neighbours of each vertex therefore gives \[ \sum_{v}\binom{d(v)}2\leq\binom n2. \] If $e$ is the number of edges, Cauchy--Schwarz gives \[ \frac{4e^2}{n}-2e\leq n(n-1). \] Solving the quadratic, \[ e\leq\frac n4\left(1+\sqrt{4n-3}\right) =\frac12n^{3/2}+\frac n4+O(\sqrt n). \tag{1} \]An explicit polarity construction
Let $q$ be an odd prime, and let the vertices be the one-dimensional subspaces of $\mathbb F_q^3$, of which there are $v=q^2+q+1$. For nonzero representatives define \[ B((x,y,z),(x',y',z'))=xz'+zx'-2yy'. \] Join two distinct projective points exactly when $B=0$. This is well-defined under scaling, symmetric, and nondegenerate, since its matrix has nonzero determinant in odd characteristic.For two distinct projective points $u,v$, the equations $B(u,w)=B(v,w)=0$ are independent linear equations: otherwise nondegeneracy would force $u,v$ proportional. Their joint solution space has dimension one. Thus the two vertices have at most one common neighbour, and the graph has no $C_4$.
The orthogonal plane to any vertex contains $q+1$ projective points. The graph degree is $q+1$, except that a self-orthogonal point loses its loop and has degree $q$. Self-orthogonality is $xz=y^2$. Its projective solutions are \[ [t^2:t:1]\quad(t\in\mathbb F_q),\qquad [1:0:0], \] so there are exactly $q+1$ of them. Therefore \[ e=\frac12\big((q^2+q+1)(q+1)-(q+1)\big) =\frac12q(q+1)^2. \tag{2} \] This includes the loop correction and establishes the sharp leading coefficient on these orders.
Passing to every large order
We use the classical prime number theorem only in the following explicit form: if $x\to\infty$, the largest prime $q\leq x$ satisfies $q/x\to1$. For any fixed $\eta>0$, this follows from $\pi(x)-\pi((1-\eta)x)\sim\eta x/\log x>0$.Take $x=(\sqrt{4n-3}-1)/2$ and the largest odd prime $q\leq x$. Then $q^2+q+1\leq n$ and $q=(1-o(1))\sqrt n$. Add isolated vertices to $2$ to obtain exactly $n$ vertices. Its edge count is $(1/2-o(1))n^{3/2}$, which together with $1$ proves the formula.