There are 31 houses on north street numbered from 1 to 57. Show at least two of them have consecutive numbers. I thought to use the pigeon hole principle but besides that not sure how to solve.

There is a unique subset with the maximum number of non-adjacent houses: {1,3,5,...,57}.  The number of houses (29) in this set is less than 30.
This observation solves the problem, but it also indicates that with 30 houses there are at least 2 adjacencies.  With only 1 adjacency $(i-1,i)$ you could add $+1$ to all the house numbers $\geq i$ and get a set of house numbers between 1 and 58 with no adjacencies, but there again the maximum size of set is 29, not 30.  To have 30 nonadjacent numbers we need to choose them from an interval of size at least 59.
Continuing the argument to the general case, to have $H$ houses with distinct numbers between $1$ and $n$, and with $\leq k$ adjacencies, one needs $(n+k) \leq 2H - 1.$ This is optimal; the equality can hold, and in a unique way.
