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Givenk   $$ k $$    graphsG    1    ,  …  ,    G    k$$ {G}_1,\dots, {G}_k $$    over a common vertex set of sizen   $$ n $$   , what is the maximum value of∑    i  ∈  [  k  ]    e  (    G    i    )   $$ {\sum}_{i\in \left[k\right]}e\left({G}_i\right) $$    having no “colorful” copy ofH   $$ H $$   , that is, a copy ofH   $$ H $$    containing at most one edge from eachG    i$$ {G}_i $$   ? Keevash, Saks, Sudakov, and Verstraëte denoted this number asex    k    (  n  ,  H  )   $$ {\mathrm{ex}}_k\left(n,H\right) $$    and completely determinedex    k    (  n  ,    K    r    )   $$ {\mathrm{ex}}_k\left(n,{K}_r\right) $$    for largen   $$ n $$   . In fact, they showed that, depending on the value ofk   $$ k $$   , one of the two natural constructions is always the extremal construction. Moreover, they conjectured that the same holds for every color‐critical graphs, and proved it for 3‐color‐critical graphs. They also asked to classify the graphsH   $$ H $$    that have only these two extremal constructions. We prove their conjecture for 4‐color‐critical graphs and for almost allr   $$ r $$   ‐color‐critical graphs whenr  &gt;  4   $$ r&gt;4 $$   . Moreover, we show that for every non‐color‐critical non‐bipartite graphs, none of the two natural constructions is extremal for certain values ofk   $$ k $$   .

On a rainbow extremal problem for color‐critical graphs