The Hajnal-Szemerédi theorem is a result in combinatorial mathematics that deals with the structure of large graphs. It states that for any graph with a sufficiently large number of vertices, if the graph does not contain a complete subgraph of a certain size, then it can be partitioned into a limited number of independent sets. This means that the graph can be divided into groups where no two vertices within the same group are connected. This theorem has important implications in the field of graph theory and helps in understanding the properties of graphs that avoid certain configurations. It is named after mathematicians József Hajnal and Endre Szemerédi, who contributed significantly to the study of extremal graph theory.