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Combinatorial design theory studies finite incidence patterns in which prescribed small collections occur equally often. A $t$-$(v,k,\lambda )$ combinatorial design consists of $v$ points and blocks of size $k$, with every $t$-element subset of points in exactly $\lambda$ blocks. Counting incidences between blocks and their $t$-element subsets giveswhere $b$ is the number of blocks.^[ref]

On $\{1,2,3,4\}$, take all four three-element subsets as blocks. Each pair occurs twice, so this is a $2$-$(4,3,2)$ combinatorial design; the count is $4\cdot 3=2\cdot 6$. These exact balance conditions turn questions about fair comparisons and repeated pairings into existence, construction and counting problems. The displayed divisibility condition is necessary, but does not by itself ensure that a design exists.

Dijen K. Ray-Chaudhuri works on combinatorial design theory. With Richard M. Wilson, he proved bounds linking design parameters and the permitted intersection sizes of blocks.^[ref]

The Ray-Chaudhuri–Wilson theorem bounds a uniform family of subsets when distinct members have only a limited number of intersection sizes. The proof encodes those restrictions in polynomials and uses linear independence to compare the family size with a finite-dimensional function space. Algebra thereby constrains a purely combinatorial configuration.^[ref]

Richard M. Wilson studies combinatorial design theory. His joint paper with Dijen K. Ray-Chaudhuri develops inequalities for designs and uniform set families.^[ref]

The Ray-Chaudhuri–Wilson theorem makes restricted intersection sizes impose an upper bound on the number of blocks. Associate a polynomial to each block, prove the resulting functions are linearly independent, and bound their number by the dimension of the ambient space. The method exemplifies the use of linear algebra in extremal combinatorics.^[ref]