Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing \(K_{s,t}^\prime \) for the subdivision of the bipartite graph K_s,t, we show that \({\rm{ex}}(n,K_{s,t}^\prime ) = O({n^{3/2 - {1 \over {2s}}}})\). This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for t sufficiently large in terms of s. Second, for any integers s,k ≥ 1, we show that \({\rm{ex}}(n,L) = \Theta ({n^{1 + {s \over <Stack><Subscript>+ 1</Subscript></Stack>}}})\) for a particular graph L depending on s and k, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number r ∈ (1, 2) is realisable in the sense that ex(n, H) = Θ(n^r) for some appropriate graph H, giving infinitely many new realisable exponents and implying that 1 + 1/k is a limit point of realisable exponents for all k ≥ 1. Writing H^k for the k-subdivision of a graph H, this result also implies that for any bipartite graph H and any k, there exists δ > 0 such that ex(n, H^k−1) = O(n^1+1/k−δ), partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph H with maximum degree r on one side which does not contain C₄ as a subgraph satisfies ex(n, H) = o(n^2−1/r).

给定图H，极值数 ex( n, H ) 是n个顶点上的无H图中的最大边数。我们在关于二部图的极值数的许多猜想上取得了进展。首先，写\(K_{s,t}^\prime \)为二部图K _s,t的细分，我们证明\({\rm{ex}}(n,K_{s,t}^ \prime ) = O({n^{3/2 - {1 \over {2s}}}})\)。这证明了 Kang、Kim 和 Liu 的猜想，并且对于t就s而言足够大的隐含常数很严格。其次，对于任何整数s,k ≥ 1，我们证明\({\rm{ex}}(n,L) = \Theta ({n^{1 + {s \over <Stack><Subscript>+ 1</Subscript></Stack>}}})\)对于取决于s和k的特定图L，回答 Kang、Kim 和 Liu 的另一个问题。这个结果涉及到 Erdős 和 Simonovits 的一个旧猜想，该猜想断言每个有理数r ∈ (1, 2) 都是可实现的，因为对于一些合适的图H，ex( n, H ) = Θ ( n ^r ) ，给出无限多个新的可实现指数并暗示 1 + 1/ k是所有k ≥ 1的可实现指数的极限点。 写作H ^k对于图H的k细分，该结果还暗示对于任何二部图H和任何k，存在δ > 0 使得 ex( n, H ^{k -1} ) = O ( n ^{1+1/ k - δ} )，部分解决了 Conlon 和 Lee 的问题。第三，扩展 Conlon 和 Lee 的最新结果，我们证明任何在一侧具有最大度数 r 且不包含C ₄作为子图的二部图H满足 ex( n, H ) = o ( n ^{2−1/ r}）。