Proofs from the Book

algebra Amer assume bijection binomial coefficients Borsuk–Ulam theorem cardinal cards cells Chapter chromatic number coefficient color column combinatorial complete complete bipartite graph complex numbers configuration conjecture consider contains convex corresponding countable defined definition denote diagonal dihedral angles dual graph elements equal equation Euler's criterion Euler’s formula example field figure filled find finite sets first fixed function geometric graph G graph theory hence implies induced subgraph induction inequality infinite integer intersect Kneser graphs Latin square Lemma length linear Math mathematics matrix modp multiple n-set number of edges obtain odd number ordinal number pair partition Paul Erd˝os permutations plane graph points polygon polyhedra polynomial polytopes probability problem proof prove quadratic residue random rectangle result riffle shuffles roots satisfies segment sequence solution subgraph subset summands Suppose Sylvester–Gallai theorem theorem tree triangle values vectors vertex set vertices yields