Proves ex(n, K_{a,b}, K_{3,b+1}) = Θ_{a,b}(n^3) for odd b ≥ 5 and 3 < a ≤ b via a projective geometry construction over finite fields.
$K_{2,t+1}$-free graphs with many copies of $K_{t,t}$
3 Pith papers cite this work. Polarity classification is still indexing.
3
Pith papers citing it
abstract
For every fixed integer $t\geq 3$, we construct an $n$-vertex $K_{2,t+1}$-free graph containing $\Omega_t(n^2)$ copies of $K_{t,t}$. Combined with a simple counting argument, this shows that \[ \mathrm{ex}(n,K_{t,t},K_{2,t+1})=\Theta_t(n^2). \] This answers a question of Spiro.
fields
math.CO 3years
2026 3verdicts
UNVERDICTED 3representative citing papers
Proves ex(n, K_{a,b}, K_{s,t}) = Theta(n^s) for s in {2,3} with s < a <= b and t large, plus existence of infinitely many r with ex(n, F, H) = Theta(n^r) for any edge-containing F.
For prime power t and n = t^{2e-1}, ex(n, K_{t,t}, K_{2,t+1}) = (1 + o(1)) n² / (2t(t-1)).
citing papers explorer
-
On the generalized Tur\'{a}n number of the complete bipartite graph $K_{3,b+1}$
Proves ex(n, K_{a,b}, K_{3,b+1}) = Θ_{a,b}(n^3) for odd b ≥ 5 and 3 < a ≤ b via a projective geometry construction over finite fields.
-
On the generalized Tur\'an number of complete bipartite graphs
Proves ex(n, K_{a,b}, K_{s,t}) = Theta(n^s) for s in {2,3} with s < a <= b and t large, plus existence of infinitely many r with ex(n, F, H) = Theta(n^r) for any edge-containing F.
-
$K_{2, t+1}$-free graphs containing an optimal number of $K_{t, t}$'s
For prime power t and n = t^{2e-1}, ex(n, K_{t,t}, K_{2,t+1}) = (1 + o(1)) n² / (2t(t-1)).