Use the pigeonhole principle to solve each of the following problems: 4. Let V = {V1, ..., V*} be any set of vectors in R2. Suppose n agents each start at (0,0) and each takes a mV-walk where a mV-walk consists of a sequence of exactly m steps and each step moves the agent along a vector in V. Prove that, if n > (M ), then some pair of agents finishes their walk at the same location. 5. Let S be a k-element subset of {1,...,n}. Prove that, if k >[n/2], then there exists X,Y ES such that x - y=1. 6. Let S be a k-element subset of {1,...,n}. Prove that, if (5) >n-1, then there exists a, b, X, Y ES such that a 7b, {a,b} + {x,y} and b - a=y- 2. (Note that it may be the case that b = x or a = y.) Use the pigeonhole principle to solve each of the following problems: 4. Let V = {V1, ..., V*} be any set of vectors in R2. Suppose n agents each start at (0,0) and each takes a mV-walk where a mV-walk consists of a sequence of exactly m steps and each step moves the agent along a vector in V. Prove that, if n > (M ), then some pair of agents finishes their walk at the same location. 5. Let S be a k-element subset of {1,...,n}. Prove that, if k >[n/2], then there exists X,Y ES such that x - y=1. 6. Let S be a k-element subset of {1,...,n}. Prove that, if (5) >n-1, then there exists a, b, X, Y ES such that a 7b, {a,b} + {x,y} and b - a=y- 2. (Note that it may be the case that b = x or a = y.)