Exact Turán numbers for even cycles

Determine the exact Turán numbers for even cycles, including the cases of the 4-cycle and the 6-cycle, whose exact values remain unresolved.

Background

The paper places the hexagon problem within the broader study of Turán numbers for even cycles. Although general upper bounds of order n^{1+1/k} are known for ex(n,C_{2k}), matching exact formulas are substantially more difficult than for odd cycles. The authors explicitly state that the exact Turán number is unresolved even for C_4 and C_6.

For C_6 specifically, the paper improves the best known upper bound to ex(n,C_6) ≤ (0.614314… + o(1))n^{4/3}, while the best known lower bound remains approximately 0.5338n^{4/3}. Thus, the exact value and leading asymptotic constant are not determined.