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# Directed graphs without rainbow stars+ The work of the first author was supported by the National Research, Development and Innovation Office – NKFIH under the grants FK 132060 and KKP-133819. The work of the second and the fourth author was supported by the National Science Centre grant 2021/42/E/ST1/00193. The work of the third author was supported by a grant from the Simons Foundation #712036.Daniel GerbnerAlfred Renyi Institute of Mathematics, HUN-REN. E-mail: gerbner.daniel@renyi.hu.Andrzej GrzesikFaculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Krakow, Poland. E-mail: Andrzej.Grzesik@uj.edu.pl.Cory PalmerDepartment of Mathematical Sciences, University of Montana. E-mail: Cory.palmer@umontana.edu.Magdalena ProrokAGH University of Krakow, al. Mickiewicza 30, 30-059 Krakow, Poland. E-mail: prorok@agh.edu.pl.Such a rainbow Turan problem was also considered for directed graphs in [3], where the optimal solution (up to a lower order error term) for \(\min_{1\leq i\leq c}e(G_{i})\) and \(\sum_{1\leq i\leq c}e(G_{i})\) was provided, for any number of colors, when a directed or transitive rainbow triangle is forbidden. Here, we continue this line of research on directed graphs and consider a directed star as the forbidden rainbow graph.Let \(S_{p,q}\) be the orientation of a star on \(p+q+1\) vertices with _center_ vertex of indegree \(p\) and outdegree \(q\). Forbidding a rainbow \(S_{p,q}\) in a collection of graphs \(\mathcal{G}=(G_{1},\ldots,G_{c})\) is analogous to forbidding a rainbow \(S_{q,p}\) in the collection of graphs obtained by changing the orientation of every edge in each graph from \(\mathcal{G}\). Thus it is enough to consider this rainbow Turan problem for \(S_{p,q}\) only when \(p\leq q\). As this problem is trivial when the number of colors \(c\) is less than the number of edges in the forbidden rainbow graph, we consider only \(c\geq p+q\).In Section 2 we consider a star \(S_{0,q}\) as the forbidden rainbow graph and prove, for every \(n>c\geq q\geq 1\), exact bounds for \(\sum_{i=1}^{c}e(G_{i})\) () and \(\min_{1\leq i\leq c}e(G_{i})\) (). In Section 3 we consider \(S_{p,q}\) as the forbidden rainbow graph for any \(q\geq p\geq 1\) and prove bounds for \(\sum_{i=1}^{c}e(G_{i})\) () and \(\min_{1\leq i\leq c}e(G_{i})\) (), which are tight up to a lower order error term. Additionally, in Section 4 we provide exact bounds for any \(c\geq 2\) and \(n\geq 3\) when a rainbow \(S_{1,1}\), i.e., directed path of length \(2\), is forbidden, for both the sum () and the minimum () of the number of edges.图片描述

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