combined the Erdös and Meshalkin theorems by adapting Meshalkin's proof of his generalized LYM inequality. They showed that the largest size of a family of p-compositions such that the sets in the i-th position of the p-tuples, ignoring duplications, are r-chain-free, for every (but not necessarily for i = p), is not greater than the sum of the largest p-multinomial coefficients. Projective geometry analog In the finite projective geometry PG(d, Fq) of dimension d over a finite field of order q, let be the family of all subspaces. When partially ordered by set inclusion, this family is a lattice. proved that the largest size of an antichain in is the largest Gaussian coefficient this is the projective-geometry analog, or q-analog, of Sperner's theorem. They further proved that the largest size of an r-chain-free family in is the sum of the r largest Gaussian coefficients. Their proof is by a projective analog of the LYM inequality. No long chains in p-compositions of a projective space obtained a Meshalkin-like generalization of the Rota–Harper theorem. In PG(d, Fq), a Meshalkin sequence of length p is a sequence of projective subspaces such that no proper subspace of PG(d, Fq) contains them all and their dimensions sum to . The theorem is that a family of Meshalkin sequences of length p in PG(d, Fq), such that the subspaces appearing in position i of the sequences contain no chain of length r for each is not more than the sum of the largest of the quantities where (in which we assume that ) denotes the p-Gaussian coefficient and the second elementary symmetric function of the numbers See also Dilworth's theorem Erdős–Ko–Rado theorem References . . . . . . . . External links Sperner's Theorem at cut-the-knot Sperner's theorem on the polymath1 wiki Families of sets Factorial and binomial topics Articles containing proofs