Let $\mathcal{H}(V,E)$ be a finite hypergraph with vertex set $V$ and edge set $E$
such that every hyperedge is non-empty. Two disjoint sets $A,B\subset V$ form a transversal
coalition in $\mathcal{H}$, if none of them is a transversal, but their union $A\cup
B$ is a transversal. A vertex partition $\Psi=\{V_1,V_2,\dots,V_p\}$ is a transversal
coalition partition, if none of the partition classes is a transversal, meanwhile
for every $i\in\{1,2,\dots,p\}$ there exists a distinct $j\in\{1,2,\dots,p\}$ such
that $V_i$ and $V_j$ form a transversal coalition. The maximum cardinality of a transversal
coalition partition of $\mathcal{H}$ is the transversal coalition number of $\mathcal{H}$
and denoted by $C_{\tau}(\mathcal{H})$. We generalize the previous upper bounds of
Barát and Blázsik on the total coalition number by using the open neighborhood hypergraph
construction. This connection was recently pointed out by Henning and Yeo and they
proved similar upper bounds on the transversal coalition number for $k$-uniform hypergraphs.
We also generalize their results by omitting the uniformity constraint. We give upper
bounds in terms of the minimum and maximum size of the hyperedges. We further investigate
this optimal case and study the transversal coalition graph. We prove that the possible
optimal transversal coalition graphs are exactly the same as the optimal total coalition
graphs.We show that every graph can be realised as a transversal coalition graph.
