In one of the hotels of the wellness planet Oxys, there are $2019$ saunas. The managers have decided to accommodate $k$ couples for the upcoming long weekend. We know the following about the guests: if two women know each other then their husbands also know each other, and vice versa. There are several restrictions on the usage of saunas. Each sauna can be used by either men only, or women only (but there is no limit on the number of people using a sauna at once, as long as they are of a single gender). Each woman is only willing to share a sauna with women whom she knows, and each man is only willing to share a sauna with men whom he does not know. What is the greatest possible $k$ for which we can guarantee, without knowing the exact relationships between the couples, that all the guests can use the saunas simultaneously while respecting the restrictions above?
1. **Understanding the Problem:**
   - We have 2019 saunas.
   - Each sauna can be used by either men only or women only.
   - Each woman is only willing to share a sauna with women she knows.
   - Each man is only willing to share a sauna with men he does not know.
   - We need to find the maximum number of couples \( k \) such that all guests can use the saunas simultaneously.

2. **Analyzing the Constraints:**
   - If two women know each other, their husbands also know each other, and vice versa.
   - Women can only share saunas with women they know.
   - Men can only share saunas with men they do not know.

3. **Optimal Assumption:**
   - Assume every woman knows every other woman. This is the worst-case scenario for women sharing saunas.
   - This implies that all women can share a single sauna.

4. **Allocation of Saunas:**
   - If all women share one sauna, we use 1 sauna for all women.
   - This leaves \( 2019 - 1 = 2018 \) saunas for the men.

5. **Men's Sauna Allocation:**
   - Since each man can only share a sauna with men he does not know, in the worst-case scenario, each man must have his own sauna.
   - Therefore, we can accommodate 2018 men in 2018 saunas.

6. **Conclusion:**
   - Each couple consists of one man and one woman.
   - We can accommodate 2018 couples, as the 2018 men will use 2018 saunas, and all women will use 1 sauna.

The final answer is \(\boxed{2018}\).