A set $S \subseteq \mathbb{Z}/p\mathbb{Z}$ is called a Sidon set if given $a, b, c, d \in S$ and $a+ b = c+ d$, then $\{a, b\} = \{c,d\}$. I was interested in knowing about the largest possible Sidon set $\mathbb{Z}/p\mathbb{Z}$ has (Here $p$ is prime). What are known about the largest possible Sidon sets contained in $\mathbb{Z}/p\mathbb{Z}$? Are there any explicit examples or constructive proof for `large' Sidon sets? Any information is appreciated. 



Sidon sets of $\mathbb{Z}/p\mathbb{Z}$
I am not sure of your level of knowledge of Sidon sets, but a good reference is O' Bryant's [survey](https://arxiv.org/pdf/math/0407117.pdf) (see section 4.3 in particular) which contains several constructions in the integers. If you are already aware of the three constructions then as far as I know, we do not know much more. We do know if you replace prime with a number of the form $n = p^2 - 1$, then the largest Sidon set of $\mathbb{Z}/n\mathbb{Z}$ has size $(1 + o(1))\sqrt{n}$, thanks to Bose and Chowla (see the cited survey for details).


We can always make a large Sidon subset of $\mathbb{Z}/p\mathbb{Z}$ from one in the integers as follows: Given $N < p/2$ and a Sidon Set 
$$A \subset\{1 , \ldots , N\},$$
 embed $A$ into $\mathbb{Z}/p\mathbb{Z}$ via the projection map. With the constructions in the aforementioned survey this gives a Sidon subset of $\mathbb{Z}/p\mathbb{Z}$ of size $(1 + o(1))\sqrt{p/2}$.


On the other hand, there is an easy counting that gives a Sidon subset of $\mathbb{Z}/p\mathbb{Z}$ has size at most $(1 + o(1))\sqrt{p}$. 


Thus the largest Sidon subset, $A$, of $\mathbb{Z}/p\mathbb{Z}$ has size
$$(1 + o(1))\sqrt{p/2} \leq |A| \leq (1 + o(1))\sqrt{p}.$$ Closing the gap in the constant is an interesting question. 


