Given a graph G on k edges, we consider the following two extremal problems: provided n is large enough, what is the minimum integer bal(n,G), if it exists, such that any 2‐coloring of the edges of a complete graph on n vertices having more than bal(n,G) edges in each color class, contains a balanced copy of G, that is, a copy of G with exactly ⌊k∕2⌋ edges in one of the colors? Graphs for which this is possible are called balanceable. The second problem deals with a similar question but we seek to guarantee copies of G in every tone, that is, having exactly r edges in, say, color red, for every 0≤r≤k. Graphs with this property are called omnitonal. We study these problems for different graph families, including paths, stars, and trees in general. When studying such extremal parameters, the question of its existence is obliged. In this line, two universal unavoidable patterns in 2‐edge‐colorings of the complete graph with sufficient representation in each of the colors emerge naturally and they are the key to characterizing balanceable as well as omnitonal graphs. For the two universal unavoidable patterns, which were already known to exist via a Ramsey‐theoretic approach, we present here a Turán‐type counterpart.

Let G be a graph with e(G) edges. We say that G is omnitonal if for every sufficiently large n there exists a minimum integer ot(n, G) such that the following holds true: For any 2-coloring f : E(K n ) → {red, blue} with more than 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, where m = m(G) depends only on G. We also present a class of graphs for which ot(n, G) = ex(n, G), the celebrated Turán numbers. Many more results and problems of similar flavor are presented.

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Abstract. For an integer d ≥ 2 which is not a square, we show that there is at most one value of the positive integer X participating in the Pell equation X 2 − dY 2 = ±1 which is a Tribonacci number, with a few exceptions that we completely characterize.

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