Motivated by a problem in theoretical computer science suggested by
Wigderson, Alon and Ben-Eliezer studied the following extremal problem
systematically one decade ago. Given a graph H, let C(n,H) be the minimum
number k such that the following holds. There are n colorings of E(Kn)
with k colors, each associated with one of the vertices of Kn, such that
for every copy T of H in Kn, at least one of the colorings that are
associated with V(T) assigns distinct colors to all the edges of E(T). In
this paper, we obtain several new results in this problem including:
\begin{itemize}
\item For paths of short length, we show that
C(n,P4)=Ω(n51) and C(n,Pt)=Ω(n31)
with t∈{5,6}, which significantly improve the previously known lower
bounds (logn)Ω(1).
\item We make progress on the problem of Alon and Ben-Eliezer about complete
graphs, more precisely, we show that C(n,Kr)=Ω(n32) when
r⩾8. This provides the first instance of graph for which the lower
bound goes beyond the natural barrier Ω(n21). Moreover, we
prove that C(n,Ks,t)=Ω(n32) for t⩾s⩾7.
\item When H is a star with at least 4 leaves, a matching of size at
least 4, or a path of length at least 7, we give the new lower bound for
C(n,H). We also show that for any graph H with at least 6 edges, C(n,H)
is polynomial in n. All of these improve the corresponding results obtained
by Alon and Ben-Eliezer.Comment: 19 page