János Komlós (mathematician)

• Komlós' theorem: He proved that every L¹-bounded sequence of real functions contains a subsequence such that the arithmetic means of all its subsequences converge pointwise almost everywhere. In probabilistic terminology, the theorem is as follows. Let ξ₁,ξ₂,... be a sequence of random variables such that E[ξ₁],E[ξ₂],... is bounded. Then there exist a subsequence ξ'₁, ξ'₂,... and a random variable β such that for each further subsequence η₁,η₂,... of ξ'₀, ξ'₁,... we have (η₁+...+η_n)/n → β a.s.
• With Miklós Ajtai and Endre Szemerédi he proved^[3] the ct²/log t upper bound for the Ramsey number R(3,t). The corresponding lower bound was established by Jeong Han Kim only in 1995, and this result earned him a Fulkerson Prize.

• The same team of authors developed the optimal Ajtai–Komlós–Szemerédi sorting network.^[4]
• Komlós and Szemerédi proved that if G is a random graph on n vertices with

${\displaystyle {\frac {1}{2}}n\log n+{\frac {1}{2}}n\log \log n+cn}$ 

edges, where c is a fixed real number, then the probability that G has a Hamiltonian circuit converges to

${\displaystyle e^{-e^{-2c}}.}$ 

• With Gábor Sárközy and Endre Szemerédi he proved the so-called blow-up lemma which claims that the regular pairs in Szemerédi's regularity lemma are similar to complete bipartite graphs when considering the embedding of graphs with bounded degrees.^[5]
• Komlós worked on Heilbronn's problem; he, János Pintz and Szemerédi disproved Heilbronn's conjecture.^[6]
• Komlós also wrote highly cited papers on sums of random variables,^[7] space-efficient representations of sparse sets,^[8] random matrices,^[9] the Szemerédi regularity lemma,^[10] and derandomization.^[11]

Komlós received his Ph.D. in 1967 from Eötvös Loránd University under the supervision of Alfréd Rényi.^[12] In 1975, he received the Alfréd Rényi Prize, a prize established for researchers of the Alfréd Rényi Institute of Mathematics. In 1998, he was elected as an external member to the Hungarian Academy of Sciences.^[13]

• Komlós–Major–Tusnády approximation