Journal of Combinatorial Theory Series A
Concentration of non-Lipschitz functions and applications
Random Structures & Algorithms - Probabilistic methods in combinatorial optimization
Random Structures & Algorithms
Divide and conquer martingales and the number of triangles in a random graph
Random Structures & Algorithms
Online balanced graph avoidance games
European Journal of Combinatorics
Sub-gaussian tails for the number of triangles in g(n, p)
Combinatorics, Probability and Computing
Subgraphs of Weakly Quasi-Random Oriented Graphs
SIAM Journal on Discrete Mathematics
The missing log in large deviations for triangle counts
Random Structures & Algorithms
Random Structures & Algorithms
A concentration result with application to subgraph count
Random Structures & Algorithms
Tight upper tail bounds for cliques
Random Structures & Algorithms
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Fix a small graph H and let YH denote the number of copies of H in the random graph G(n, p). We investigate the degree of concentration of YH around its mean, motivated by the following questions.• What is the upper tail probability Pr(YH ≥ (1 + ε)𝔼(YH))?• For which λ does YH have sub-Gaussian behaviour, namely***** Insert formula here *****where c is a positive constant?• Fixing λ = ω(1) in advance, find a reasonably small tail T = T(λ) such that***** Insert formula here *****We prove a general concentration result which contains a partial answer to each of these questions. The heart of the proof is a new martingale inequality, due to J. H. Kim and the present author [13].