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Author Archives: Anurag Bishnoi
Extending Ryser’s conjecture
In an earlier post, I talked about Ryser’s conjecture on partite uniform hypergraphs, that has stayed open for all despite a considerable effort by several mathematicians over a period of 50 years. A bit more effort has been spent on … Continue reading
Posted in Combinatorics, Extremal Combinatorics
Tagged hypergraphs, Patrick Morris, postdoc, research, Ryser, Shagnik Das, Tibor Szabo, vertex cover
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Bound on Ramsey numbers from finite geometry
In an earlier post I talked about the work of Mubayi and Verstraete on determining the offdiagonal Ramsey numbers via certain optimal pseudorandom graphs, which are not yet known to exist except for the case of triangles. Beyond this conditional … Continue reading
Kopparty’s graph
Alon’s construction of optimal pseudorandom graphs from 1994 is useful for obtaining several interesting combinatorial results in various areas of mathematics, some of which are highlighted in this survey of Noga from last year (also see my previous post). In … Continue reading
Ramsey numbers from pseudorandom graphs
One of the foundational results in modern combinatorics is Ramsey’s theorem which states that for every positive integers there exists a constant such that for all , every coloring of edges of the complete graph has a monochromatic copy of … Continue reading
Spectral proofs of theorems on the boolean hypercube
In a recent breakthrough Hao Huang proved the sensitivity conjecture, that had remained open for past 30 years despite some serious effort from various computer scientists and mathematicians. The proof can be described in a single tweet (if you have … Continue reading
Pseudorandom cliquefree graphs
Pseudorandom graphs are graphs that in some way behaves like a random graph with the same edge density. One way in which this happens is as follows. In the random graph , with , a direct application of Chernoff bound … Continue reading
Posted in Combinatorics, Extremal Combinatorics, Spectral Graph Theory
Tagged alon, pseudorandom, pseudorandom graphs
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Ryser’s conjecture
I am on a research visit in Rome, working with Valentina Pepe, and our joint paper on Ryser’s conjecture is on arXiv now. So this seems like the right time to talk about the conjecture and the problems related to … Continue reading