  This paper gives tight upper bounds on the number of edges and the index for
$\mathcal{K}^-_{r + 1}$-free unbalanced signed graphs, where $\mathcal{K}^-_{r
+ 1}$ is the set of $r+1$-vertices unbalanced signed complete graphs.
  \indent We first prove that if $\Gamma$ is an $n$-vertices $\mathcal{K}^-_{r
+ 1}$-free unbalanced signed graph, then the number of edges of $\Gamma$ is
$$e(\Gamma) \leq \frac{n(n-1)}{2} - (n - r ).$$
  \indent Let $\Gamma_{1,r-2}$ be a signed graph obtained by adding one
negative edge and $r - 2$ positive edges between a vertex and an all positive
signed complete graph $K_{n - 1}$.
  Secondly, we show that if $\Gamma$ is an $n$-vertices $\mathcal{K}^-_{r +
1}$-free unbalanced signed graph, then the index of $\Gamma$ is
  $$\lambda_{1}(\Gamma) \leq \lambda_{1}(\Gamma_{1,r-2}), $$ with equality
holding if and only if $\Gamma$ is switching equivalent to $\Gamma_{1,r-2}$.
  \indent It is shown that these results are significant in extremal graph
theory.
  Because they can be regarded as extensions of Tur{\'a}n's Theorem [Math. Fiz.
Lapok 48 (1941) 436--452] and spectral Tur{\'a}n problem [Linear Algebra Appl.
428 (2008) 1492--1498] on signed graphs, respectively.
  Furthermore, the second result partly resolves a recent open problem raised
by Wang [arXiv preprint arXiv:2309.15434 (2023)].

Combinatorics