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The extremal number of the subdivisions of the complete bipartite graph
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abstract
For a graph $F$, the $k$-subdivision of $F$, denoted $F^k$, is the graph obtained by replacing the edges of $F$ with internally vertex-disjoint paths of length $k$. In this paper, we prove that $\mathrm{ex}(n,K_{s,t}^k)=O(n^{1+\frac{s-1}{sk}})$, which is tight for $t$ sufficiently large. This settles a conjecture of Conlon--Janzer--Lee, and improves on a substantial body of work by Conlon--Janzer--Lee and Jiang--Qiu.
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Many Turan exponents via subdivisions
Every rational number 1 + p/q with q > p^2 is shown to be the exact growth exponent of some bipartite Turan problem.
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