Finding a seating arrangement of $4$ different people in $n$ rounds I have a practical problem. I want to arrange a speeddate event with $24$ or $32$ people in $7$ rounds. I have a room with sufficiently many tables and I want $4$ people per table and each round must consist of $4$ different people per table.

Is it possible to find such an arrangement?
It is impossible for 24 and 32 people. I'll demonstrate how it is impossible for 24 people (the case for 32 is similar)

Suppose there are 24 people. This means that every round there will be 6 tables of 4. In total, there are 24C2 ways to choose a pair of people from this group of 24, and every such pair must have sat together at a table in one of the rounds. Each table contributes a maximum of 4C2 new pairs, so we have the inequality

4C2 * 6 * 7 >= 24C2

which evaluates to be 252 >= 276, which is a contradiction.

What do you wish to keep constant? Do you want a speed dating with exactly 7 rounds? Or do you want the minimum number of rounds for 24 or 32 people?
It is impossible for 24 and 32 people. I'll demonstrate how it is impossible for 24 people (the case for 32 is similar)

Suppose there are 24 people. This means that every round there will be 6 tables of 4. In total, there are 24C2 ways to choose a pair of people from this group of 24, and every such pair must have sat together at a table in one of the rounds. Each table contributes a maximum of 4C2 new pairs, so we have the inequality

4C2 * 6 * 7 >= 24C2

which evaluates to be 252 >= 276, which is a contradiction.

What do you wish to keep constant? Do you want a speed dating with exactly 7 rounds? Or do you want the minimum number of rounds for 24 or 32 people?