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Register a new userA Ramsey variant of the Brown-ErdH{o}s-Sos conjecture
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An $r$-uniform hypergraph ($r$-graph for short) is called linear if every pair of vertices belong to at most one edge. A linear $r$-graph is complete if every pair of vertices are in exactly one edge. The famous Brown-ErdH{o}s-Sos conjecture states that for every fixed $k$ and $r$, every linear $r$-graph with $Omega(n^2)$ edges contains $k$ edges spanned by at most $(r-2)k+3$ vertices. As an intermediate step towards this conjecture, Conlon and Nenadov recently suggested to prove its natural Ramsey relaxation. Namely, that for every fixed $k$, $r$ and $c$, in every $c$-colouring of a complete linear $r$-graph, one can find $k$ monochromatic edges spanned by at most $(r-2)k+3$ vertices. We prove that this Ramsey version of the conjecture holds under the additional assumption that $r geq r_0(c)$, and we show that for $c=2$ it holds for all $rgeq 4$.
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We prove the well-known Brown-ErdH{o}s-Sos Conjecture for hypergraphs of large uniformity in the following form: any dense linear $r$-graph $G$ has $k$ edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$ is large enough given the linear density of $G$, and the number of vertices of $G$ is large enough given $r$ and $k$.
The ErdH{o}s-Faber-Lov{a}sz conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stabili
Extending the concept of Ramsey numbers, Erd{H o}s and Rogers introduced the following function. For given integers $2le s<t$ let $$ f_{s,t}(n)=min {max {|W| : Wsubseteq V(G) {and} G[W] {contains no} K_s} }, $$ where the minimum is taken over all $K_t$-free graphs $G$ of order $n$. In this paper, we show that for every $sge 3$ there exist constants $c_1=c_1(s)$ and $c_2=c_2(s)$ such that $f_{s,s+1}(n) le c_1 (log n)^{c_2} sqrt{n}$. This result is best possible up to a polylogarithmic factor. We also show for all $t-2 geq s geq 4$, there exists a constant $c_3$ such that $f_{s,t}(n) le c_3 sqrt{n}$. In doing so, we partially answer a question of ErdH{o}s by showing that $lim_{nto infty} frac{f_{s+1,s+2}(n)}{f_{s,s+2}(n)}=infty$ for any $sge 4$.
A graph is $P_8$-free if it contains no induced subgraph isomorphic to the path $P_8$ on eight vertices. In 1995, ErdH{o}s and Gy{a}rf{a}s conjectured that every graph of minimum degree at least three contains a cycle whose length is a power of two. In this paper, we confirm the conjecture for $P_8$-free graphs by showing that there exists a cycle of length four or eight in every $P_8$-free graph with minimum degree at least three.
For a 2-connected graph $G$ on $n$ vertices and two vertices $x,yin V(G)$, we prove that there is an $(x,y)$-path of length at least $k$ if there are at least $frac{n-1}{2}$ vertices in $V(G)backslash {x,y}$ of degree at least $k$. This strengthens a well-known theorem due to ErdH{o}s and Gallai in 1959. As the first application of this result, we show that a 2-connected graph with $n$ vertices contains a cycle of length at least $2k$ if it has at least $frac{n}{2}+k$ vertices of degree at least $k$. This confirms a 1975 conjecture made by Woodall. As another applications, we obtain some results which generalize previous theorems of Dirac, ErdH{o}s-Gallai, Bondy, and Fujisawa et al., present short proofs of the path case of Loebl-Koml{o}s-S{o}s Conjecture which was verified by Bazgan et al. and of a conjecture of Bondy on longest cycles (for large graphs) which was confirmed by Fraisse and Fournier, and make progress on a conjecture of Bermond.
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