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 containing an optimal number of $K_{t, t}$'s
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
The generalized Tur\'{a}n number $ex(n, K_{t, t}, K_{2, t+1})$ is the maximum number of copies of $K_{t, t}$ that a $K_{2, t+1}$-free graph on $n$ vertices can contain. Recently, Pohoata, Tidor, and Yu established that $ex(n, K_{t, t}, K_{2, t+1}) = \Theta_t(n^2)$ for all integers $t \geq 3$. In this short note, we use an explicit construction to establish that when $t$ is a prime power and $n = t^{2e - 1}$, then $$ ex(n, K_{t, t}, K_{2, t+1}) = (1 + o(1))\frac{n^2}{2t(t-1)}. $$
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math.CO 2years
2026 2verdicts
UNVERDICTED 2representative 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.
citing papers explorer
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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.
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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.