Tag Archives: combinatorics

Wenger graphs

A central (and foundational) question in extremal graph theory is the forbidden subgraph problem of Turán, which asks for the largest number of edges in an -vertex graph that does not contain any copy of a given graph as its … Continue reading

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The footprint bound

Studying the set of common zeros of systems of polynomial equations is a fundamental problem in algebra and geometry. In this post we will look at estimating the cardinality of the set of common zeros, when we already know that … Continue reading

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Introduction to polynomial method

(The following is a blog-friendly version of Chapter 7 of my PhD thesis, which is an introduction to the so-called polynomial method.) The polynomial method is an umbrella term for different techniques involving polynomials which have been used to solve … Continue reading

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What I have learned in finite geometry

On September 2nd, 2014 I wrote a blog post titled learning finite geometry, in which I described how much I have learned in my first year of PhD and more importantly, the topics that I wish to learn while I … Continue reading

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Expander Mixing Lemma in Finite Geometry

In this post I will discuss some nice applications of the expander mixing lemma in finite incidence geometry, including a new result that I have obtained recently. In many of the applications of the lemma in finite geometry, the graph is bipartite, and … Continue reading

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Incidence Bounds and Interlacing Eigenvalues

The Szemerédi–Trotter theorem is one of the central results in discrete geometry which gives us a (tight) bound on the number of incidences, i.e., the number of point-line pairs with the point lying on the line, between finite sets of points and lines … Continue reading

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The Erdős-Ginzburg-Ziv theorem

Let be  a sequence of integers (not necessarily distinct). Then there exists a subsequence of the sum of whose elements is divisible by .  This is one of the first problems I saw when learning the pigeonhole principle. And it’s … Continue reading

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