所以我遇到了这个奇怪的错误：您无法在 Springer Latex 模板中以垂直模式使用“\unskip”。无论出于何种原因，这都是由最后一个 \end{proof} 之前的空白行引起的。也不确定如何生成较小的示例。
https://www.overleaf.com/2166494628jhkmqjhhjjsv
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\begin{document}
Let us give the details of the claim in \cite{ErdosPosaFlorent} that even cycles satisfy the Erdos Posa property with bounding function $f(k)=131k$.
Let $\mathcal{D}$ denote the class of diamonds, that is, graphs that are a subdivision of the graph on two vertices with three parallel edges between them.
Given a set $\mathcal{H}$ of graphs denote by $ \pck_\mathcal{H}(G)$ the maximum number of vertex disjoint copies of graphs in $\mathcal{H}$ and by $\cover_\mathcal{H}(G)$ the minimum size of an $\mathcal{H}$-transversal, a set of vertices $X$ for which $G \backslash X$ contains no subgraph isomorphic to a graph in $\mathcal{H}$.
Let $\trw(G)$ denote the treewidth of $G$.
For $x \in \mathbb{R}$, denote by $\lceil x \rceil $ smallest integer larger than $x$.
%%Denote by $\mathcal{D}$ the set of diamonds
That even cycles satisfy Erdos-Posa with $f(k)= 131k$ follows as a simple consequence of the following result of Fomin, Saurabh and Thilikos \cite{FominErdos}.
\begin{theorem}\label{packing diamonds}
\cite{FominErdos} For a planar graph $G$ $\pck_\mathcal{D}(G) \leq 131 \cover_\mathcal{D} (G) $.
\end{theorem}
\begin{lemma}
For a planar graph $ G $, $\trw(G) \leq 37 \pck(G)^{\frac{1}{2}}$.
\end{lemma}
\begin{proof}
% We show $\trw(G) \leq 18 \lceil \pck_\mathcal{D} (G)^{\frac{1}{2}} \rceil +1 $.
%% We need the following result of Robertson, Seymour and Thomas.
The following lemma
\begin{lemma}\cite{SeymourGrid}
Let $g \geq 1$ be an integer.
Every planar graph with no $g \times g$ grid minor has tree-width $ \leq 6g-5 $
\end{lemma}
\end{proof}
\bibliography{temp, EvenCycleTransversal}
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\appendix
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