  The goal of the paper is to give fine-grained hardness results for the
Subgraph Isomorphism (SI) problem for fixed size induced patterns $H$, based on
the $k$-Clique hypothesis that the current best algorithms for Clique are
optimal.
  Our first main result is that for any pattern graph $H$ that is a {\em core},
the SI problem for $H$ is at least as hard as $t$-Clique, where $t$ is the size
of the largest clique minor of $H$. This improves (for cores) the previous
known results [Dalirrooyfard-Vassilevska W. STOC'20] that the SI for $H$ is at
least as hard as $k$-clique where $k$ is the size of the largest clique {\em
subgraph} in $H$, or the chromatic number of $H$ (under the Hadwiger
conjecture). For detecting \emph{any} graph pattern $H$, we further remove the
dependency of the result of [Dalirrooyfard-Vassilevska W. STOC'20] on the
Hadwiger conjecture at the cost of a sub-polynomial decrease in the lower
bound.
  The result for cores allows us to prove that the SI problem for induced
$k$-Path and $k$-Cycle is harder than previously known. Previously [Floderus et
al. Theor. CS 2015] had shown that $k$-Path and $k$-Cycle are at least as hard
to detect as a $\lfloor k/2\rfloor$-Clique. We show that they are in fact at
least as hard as $3k/4-O(1)$-Clique, improving the conditional lower bound
exponent by a factor of $3/2$.
  Finally, we provide a new conditional lower bound for detecting induced
$4$-cycles: $n^{2-o(1)}$ time is necessary even in graphs with $n$ nodes and
$O(n^{1.5})$ edges.
