We consider unavoidable chromatic patterns in $2$-colorings of the edges of the complete graph. Several such problems are explored being a junction point between Ramsey theory, extremal graph theory (Tur\'an type problems), zero-sum Ramsey theory, and interpolation theorems in graph theory. A role-model of these problems is the following: Let $G$ be a graph with $e(G)$ edges. We say that $G$ is omnitonal if there exists a function ${\rm ot}(n,G)$ such that the following holds true for $n$ sufficiently large: For any $2$-coloring $f: E(K_n) \to \{red, blue \}$ such that there are more than ${\rm ot}(n,G)$ edges from each color, and for any pair of non-negative integers $r$ and $b$ with $r+b = e(G)$, there is a copy of $G$ in $K_n$ with exactly $r$ red edges and $b$ blue edges. We give a structural characterization of omnitonal graphs from which we deduce that omnitonal graphs are, in particular, bipartite graphs, and prove further that, for an omnitonal graph $G$, ${\rm ot}(n,G) = \mathcal{O}(n^{2 - \frac{1}{m}})$, where $m = m(G)$ depends only on $G$. We also present a class of graphs for which ${\rm ot}(n,G) = ex(n,G)$, the celebrated Tur\'an numbers. Many more results and problems of similar flavor are presented.

中文翻译：

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完整图形的 2-着色中不可避免的彩色图案

我们在完整图的边缘的 $2$-colorings 中考虑不可避免的彩色图案。研究了几个这样的问题，作为图论中拉姆齐理论、极值图论（Tur\'an 类型问题）、零和拉姆齐理论和插值定理之间的连接点。这些问题的一个角色模型如下：让 $G$ 是一个带有 $e(G)$ 边的图。如果存在一个函数 ${\rm ot}(n,G)$ 使得以下对于足够大的 $n$ 成立，我们说 $G$ 是全调的： 对于任何 $2$-coloring $f: E(K_n ) \to \{red, blue \}$ 使得每种颜色有超过 ${\rm ot}(n,G)$ 边，并且对于任何一对非负整数 $r$ 和 $b$ $r+b = e(G)$，$K_n$ 中有$G$ 的副本，正好有$r$ 红边和$b$ 蓝边。我们给出了全调图的结构特征，从中我们推断全调图尤其是二部图，并进一步证明，对于全调图 $G$，${\rm ot}(n,G) = \mathcal {O}(n^{2 - \frac{1}{m}})$，其中 $m = m(G)$ 仅取决于 $G$。我们还展示了一类图，其中 ${\rm ot}(n,G) = ex(n,G)$，著名的图尔安数。提出了更多类似风味的结果和问题。