There are 2019 chairs at a round table. What is the minimum number of chairs that must be occupied such that there are some consecutive set of 4 chairs (or more) occupied.
The chairs can be divided in sets $S_{k} ={4k+1,4k+2,4k+3,4k+4}$ for $k={0,…,503}$ and $S_{504} = \{2017,2018,2019\} $. If one seat is kept open in each set (at least 505 open seats) a sequence of 4 occupied seats may not appear. So, in the worst case scenario, we still would be able to find such arrangement that 1514 people would be seated and there may not be even one quadruple of consecutive seats e.g. $$ \begin{array}{c|c|c|c|c|c} ⚪ ⚫ ⚫ ⚫&⚪ ⚫ ⚫ ⚫& ...&⚪ ⚫ ⚫ ⚫&⚪ ⚫ ⚫&⚪ ⚫ ⚫ ⚫\\ S_0 & S_1 & ... & S_{503} & S_{504}&S_0 \end{array} $$
Applying the pigeonhole principle:
filling more than $2019−505=1514$ seats, with
- at least 1513 seats filled for $S_0,…,S_{503}$ or
- 1512 seats filled for $S_0,…,S_{503}$ and 3 for $S_{504}$
guarantees us to find such 4 seats, since $1513>3*504$. In the second case if we have 3 sits occupied in the last set they will always be adjacent to an occupied seat in either $S_0$ or $S_{503}$.
This solution was insufficient, what did I miss?
I think the only gap in your solution is the claim "If we have 3 seats occupied in the last set they will always be adjacent to an occupied seat in either $S_0$ or $S_{503}$." That need not be true; the first seat of $S_0$ and the last seat of $S_{503}$ could both be empty.
You then need to prove that in this case there are four adjacent occupied seats somewhere around the table. Since 1515 seats are occupied, and three of them are in $S_{504}$, exactly three seats must be occupied in each of $S_0$ through $S_{503}$ (or else one of them has all four occupied and we are done).
If the unoccupied seat of $S_1$ is not the first one, then we have four occupied seats in a row across $S_0$ and $S_1$. Proceeding around the table, we see that if the first seat of $S_i$ is unoccupied, then the first seat of $S_{i+1}$ must be unoccupied also; but this can't continue indefinitely since we know the unoccupied set of $S_{503}$ is the last one.