Konrad J. Swanepoel coauthored an extension of Sylvester-Gallai theorem geometry with Noam Elkies and Lou M. Pretorius. Their paper gives an elementary proof that a finite Sylvester-Gallai configuration over the complex numbers lies in a projective plane, and extends the conclusion over the quaternions to a three-dimensional flat.^[ref]

The result shows that incidence constraints depend on the underlying scalars: the corresponding real configuration must already be collinear.

Alexander Sidorenko works on density inequalities in graph theory and the theory of hypergraphs. His name is associated with the Sidorenko conjecture, which compares copies of a fixed bipartite graph with the random-density benchmark.^[ref]

With David Conlon and Joonkyung Lee, he studied hypergraph failures of the analogous inequality. Their result turns such failures into improved extremal constructions, connecting analytic inequalities for densities with concrete forbidden-subgraph problems.^[ref]

Joonkyung Lee works on extremal combinatorics. With David Conlon and Alexander Sidorenko, he studied connections between the Sidorenko conjecture and extremal numbers.^[ref]

The graph conjecture concerns lower bounds on homomorphism densities for bipartite graphs. The paper investigates hypergraph analogues, where counterexamples exist, and uses those failures to improve lower bounds for the number of edges in a forbidden-configuration-free hypergraph.^[ref]

Benny Sudakov studies combinatorics, especially extremal and probabilistic questions. His ETH author page documents work in these areas.^[ref]

With David Conlon, Jacob Fox and Choongbum Lee, he developed bounds for ordered Ramsey numbers. Their work compares the order-preserving problem with the ordinary Ramsey number and demonstrates a large gap for some ordered matchings.^[ref]

Choongbum Lee works on graph theory and Ramsey theory. His joint paper with David Conlon, Jacob Fox and Benny Sudakov investigates copies of a graph whose vertex order must be preserved.^[ref]

The ordered setting is sensitive to labeling: some ordered matchings have superpolynomial Ramsey numbers. Thus low ordinary degree does not by itself make an order-constrained embedding problem easy.^[ref]

Jacob Fox works on extremal combinatorics, Ramsey theory and algebraic and probabilistic methods in graph theory. His Stanford research page also identifies applications to computer science.^[ref]

With David Conlon, Choongbum Lee and Benny Sudakov, he studied ordered Ramsey numbers. Requiring the vertices of a monochromatic copy to appear in a prescribed order can force superpolynomial growth even when the underlying graph is a matching.^[ref]

David Conlon studies extremal graph theory and Ramsey theory. His joint work with Jacob Fox, Choongbum Lee and Benny Sudakov shows that imposing a vertex order can make Ramsey problems substantially harder, even for matchings.^[ref]

With Joonkyung Lee and Alexander Sidorenko, he also linked failures of a hypergraph analogue of the Sidorenko conjecture to improved extremal lower bounds. This turns a failed density inequality into information about constructions avoiding a specified hypergraph.^[ref]

Dona Strauss works on algebraic methods in Ramsey theory. Her joint paper with Neil Hindman and Imre Leader examines when different Milliken-Taylor systems can be monochromatic together.^[ref]

Compression removes zero entries and repeated consecutive coefficients. For positive coefficient patterns, the resulting sequences must satisfy a positive-rational-proportionality criterion in the theorem's simultaneous setting. The result exposes structure behind a coloring problem whose individual patterns already satisfy strong partition properties.^[ref]

Imre Leader studies extremal combinatorics and Ramsey theory. His Cambridge research profile identifies these areas.^[ref]

With Neil Hindman and Dona Strauss, he studied compatibility between Milliken-Taylor systems. The coefficients of two systems cannot be chosen arbitrarily if both are to be monochromatic in the required simultaneous sense. Their theorem identifies a positive-rational-proportionality condition for compressed positive coefficient patterns, sharpening the distinction between individually unavoidable patterns and jointly compatible ones.^[ref]

Kendall Williams coauthored work on polynomial extensions of the Milliken–Taylor theorem with Vitaly Bergelson and Neil Hindman. The primary manuscript verifies this exact byline.^[ref]

Their results find monochromatic polynomial configurations built from separated finite blocks of a sequence. The approach combines Ramsey theory with the algebra of ultrafilters, extending structured finite-sum patterns to more general polynomial images.^[ref]

Neil Hindman works on Ramsey theory and algebraic methods for finding structured monochromatic sets. With Vitaly Bergelson and Kendall Williams, he developed polynomial extensions of the Milliken–Taylor theorem, using ultrafilters and their algebra.^[ref]

With Imre Leader and Dona Strauss, he also studied when different Milliken-Taylor systems can be monochromatic simultaneously. The coefficient patterns impose genuine compatibility conditions.^[ref]

Dömötör Pálvölgyi studies combinatorics and discrete geometry, particularly colorings of geometric hypergraphs. His ELTE research statement identifies covering decomposition as a motivating problem.^[ref]

A typical question asks whether a sufficiently deep multiple covering of the plane by unit disks can be split into two coverings. Coloring the disks with two colors reformulates the goal: every point must lie in a disk of each color. This connects geometric coverage with a hypergraph-coloring constraint, translating a continuous geometric problem into combinatorial structure.^[ref]

John R. Stembridge works on algebraic combinatorics, representation theory, Coxeter groups and symmetric functions. His Michigan profile also identifies computational methods in algebra as a focus.^[ref]

With Richard P. Stanley, he studied immanants of matrices associated with the Jacobi–Trudi identity. An immanant replaces the sign character in a determinant by another symmetric group character. Their paper proves special cases of stronger positivity conjectures and connects the calculation with cycle structure of permutations having restricted positions.^[ref]

Paul Erdős worked on number theory, combinatorics and extremal problems. His joint 1948 paper with Nicolaas Govert de Bruijn gives an incidence bound from a unique-line-through-two-points condition.^[ref]

For a finite nontrivial incidence system, the number of lines is at least the number of points. Equality forces a near-pencil or a finite projective plane. The theorem illustrates an extremal approach: determine a sharp numerical bound, then understand the configurations attaining it.^[ref]

Nicolaas Govert de Bruijn worked on combinatorics and mathematical structures. With Paul Erdős, he proved a finite incidence theorem in their 1948 paper: a nontrivial system in which each pair of points lies on exactly one line has at least as many lines as points.^[ref]

The equality cases are highly organized: a near-pencil or a finite projective plane. This shows how a local pair-incidence rule controls both a global counting inequality and the structure of extremal examples.^[ref]

Will Sawin works on problems connecting arithmetic and combinatorics. His paper on the union-closed sets conjecture applies entropy inequalities to a finite family closed under unions.^[ref]

For a nontrivial such family, the paper guarantees an element in at least a $(3-\sqrt{5})/2$ fraction of the sets. The argument studies the union of independent random members and bounds the resulting entropy growth. It is a quantitative advance toward the conjectured one-half fraction, with the nonempty-union condition needed to exclude the family containing only the empty set.^[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]

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]

Giorgis Petridis works on growth estimates in additive combinatorics. His paper gives a short proof of Plünnecke inequalities and extensions to product sets in groups.^[ref]

The proof chooses a subset with a minimal expansion ratio and uses that choice to control further sums. This converts one small-growth hypothesis into bounds for iterated sum-and-difference sets. The noncommutative setting requires separate product-set arguments because the order of multiplication matters.^[ref]

Ernie Croot and Olof Sisask introduced a sampling technique for approximate translation invariance in additive combinatorics. Their Croot-Sisask almost-periodicity theorem controls how much a convolution changes under many translations.^[ref]

The useful shift is from exact algebraic symmetry to small error in a function norm. Random samples approximate averages; comparing repeated approximations exposes almost-periods. The original paper carefully specifies counting convolution and counting $L^p$ norms, which determine the constants and scale of the assertion.^[ref]