Generalized Turán problems for even cycles

Given a graph H and a set of graphs F, let ex(n;H;F) denote the maximum possible number of copies of H in an F-free graph on n vertices. We investigate the function ex(n;H;F), when H and members of F are cycles. Let C_k denote the cycle of length k and let C_k = {C₃;C₄;…;C_k}. We highlight the main results below. (i) We show that ex(n;C_2l;C_2k) = Θ(nl) for any l; k ≥ 2. Moreover, in some cases we determine it asymptotically. (ii) Erdős's Girth Conjecture states that for any positive integer k, there exist a constant c > 0 depending only on k, and a family of graphs {G_n} such that (image found) with girth more than 2k. Solymosi and Wong proved that if this conjecture holds, then for any l ≥ 3 we have (image found) We prove that their result is sharp in the sense that forbidding any other even cycle decreases the number of C_2l's significantly. (iii) We prove (image found) provided a stronger version of Erdős's Girth Conjecture holds (which is known to be true when l = 2; 3; 5). This result is also sharp in the sense that forbidding one more cycle decreases the number of C_2l+1's significantly.