Zilin Jiang
Joint work with Boris Bukh
$\operatorname{ex}(n, F)$ – largest # of edges in an $F$-free graph
$$\operatorname{ex}(n, F) = \left(1 - \frac{1}{\chi - 1}\right){n \choose 2} + o(n^2)$$
Degenerate case $\chi = 2$
Erdős: $\operatorname{ex}(n, C_{2k}) \le$ $\gamma_k$ $n^{1+1/k}$
1974 | Bondy–Simonovits | $20k$ | |
2000 | Verstraëte | $8(k-1)$ | |
2012 | Pikhurko | $k-1$ | |
2014 | Bukh–J. | $80\sqrt{k\log k}$ |
▶ | Revisit Pikhurko's proof |
How to make more math? |
If | $V(\Theta) = $ $A$ $\cup$ $B$ is not bipartition |
Then | Path from $A$ to $B$ of any possible length |
Suppose $G$ is $C_{2k}$-free and $e(G) \ge n^{1+1/k}$
Take $H\subset G$ of minimum degree $\delta(H) \ge e(G) / n \ge n^{1/k}$
Claim | $H_i$ is $\Theta$-free for $i < k$ |
Find $y$ the rightmost common ancestor of $\Theta \cap V_i$ in $T$.
Pick $z $ a child of $y$, and find $C_{2k}$ using $\Theta$.
Obs | Bipartite graph of min deg $\ge k$ contains $\Theta$-graph. |
Cor | Average degree of $H_i \le 2k$. |
$\frac{v_{i+1}}{v_i}\lesssim \frac{\delta}{k}$$\implies \left(\frac{\delta}{k}\right)^k \lesssim n$$\implies e(G) \lesssim k\ n^{1+1/k}$
▶ | Review Pikhurko's proof |
▶ | How to make more math? |
Large minimum degree | |
Implies | Denser trilayered subgraph |
Or | $\Theta$-graph on two levels |
Question | Is $Bg$ big? |
Yes | $V_{i+1} = Bg \cup Sm$ |
No | $V_{i+1} = Sm$ |