| Summary: | This thesis studies a range of topics across combinatorics, broadly defined. The second chapter of this thesis addresses a longśstanding question of Erdős regarding the existence of high girth Steiner triple systems. The tools employed fall squarely within the context of probabilistic method, drawing on recent advances within design theory and the theory of random processes. The third and fourth chapters of this thesis consider problems within random matrix theory; in particular on problems regarding sparse random graphs. The third chapter concerns a question of Vu regarding the singularity of the k-core of a random graph. In particular, a sparse ErdősśRényi graph G(n,d/n) with high probability has large corank due to the presence of isolated vertices. Answering raised by Vu at the ICM 2014, the third chapter proves that by iteratively deleting vertices of degree less than k (e.g. forming the k-core) we have that the associated graph is nonsingular with high probability. The fourth chapter answers a longstanding question regarding the spectral distribution of a matrix where each entry is 1 with probability d/n. In particular, this result gives the first spectral distribution for nonhermitian random matrices at this level of sparsity and answers a question that was highlighted by Tikhomirov at the ICM 2022. The fifth and sixth chapters are concerned with discrepancy theory. The fifth chapter provides bounds for online vector balancing by finding a Markov chain on R with integer steps and for which the stationary distribution is Gaussian. The sixth chapter concerns a famous result of Spencer in finding a low discrepancy coloring of a set system. This chapter gives the first such algorithm for finding a low discrepancy coloring which runs in nearly input sparsity time. The seventh chapter concerns effective bounds for special cases of the polynomial Szemerédi theorem. In particular, answering a question of Green, this chapter gives effective bounds for sets avoiding the pattern x,x + y² − 1,x + 2(y² − 1) (e.g. Roth’s theorem with a shifted square difference). This is the first polynomial pattern which is not homogeneous and complexity at least one for which effective bounds have been obtained. Furthermore this paper introduces the use of higher order techniques within the context of degree-lowering. The final chapter concerns a question at the intersection of probabilistic combinatorics and statistical physics. This chapter determines the sharp constant γ such that with high 3probability a graph G ∼ G(n,1/2) may be split into two equal parts A and B such that each vertex in A has γ√n more neighbors in A than in B. This provides an essentially complete resolution to a question of Füredi and draws on a combination of methods from graph enumeration and boolean analysis.
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