Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. In 1975, Erdos raised the problem of determining $f(n)$ (see J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976), p.247, Problem 11). Given a graph $H$, what is the maximum number of edges of a graph with $n$ vertices not containing $H$ as a subgraph? This number is denoted $ex(n,H)$, and is known as the Turan number. Erdos conjectured that there exists a positive constant $c$ such that $ex(n,C_{2k})\geq cn^{1+1/k}$(see P. Erdos, Some unsolved problems in graph theory and combinatorial analysis, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) , pp. 97--109, Academic Press, London, 1971). Haj\'{o}s conjectured that every simple even graph on $n$ vertices can be decomposed into at most $n/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp. 231 - 236 ). We present the problems, conjectures related to these problems and we summarize the know results.

Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. Erd\"{o}s raised the problem of determining $f(n)$. Erd\"{o}s conjectured that there exists a positive constant $c$ such that $ex(n,C_{2k})\geq cn^{1+1/k}$. Haj\'{o}s conjecture that every simple even graph on $n$ vertices can be decomposed into at most $n/2$ cycles. We present the problems, conjectures related to these problems and we summarize the know results. We do not think Haj\'{o}s conjecture is true.

Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix 0<α<1. Let N_α(d) denote the maximum number of lines through the origin in R^d with pairwise common angle arccos α. Let k denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly (1−α)/(2α). If k<∞, then N_α(d)= ⌊ k(d−1)/(k−1) ⌋ for all sufficiently large d, and otherwise N_α(d)=d+o(d). In particular, N_{1/(2k−1)}(d)= ⌊ k(d−1)/(k−1) ⌋ for every integer k≥2 and all sufficiently large d. A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.

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Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition with k parts if there exists an ε-regular partition with k parts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.