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I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here:

For $i=1,2,\ldots,n$, let $d_i$ be a sequence of positive integers satisfying $d_1\geq\cdots\geq d_n$ with $2\leq d_i\leq n-1$, and $r_i$ a sequence of positive real numbers with $2\leq d_i\leq n-1$ that satisfy the following system of equations:

$$d_i=\sum_{j=1}^n \frac{r_ir_j}{1+r_ir_j}$$

for each $i=1,2\ldots,n$.

I have additional constraints that $d_1+\cdots+d_n$ is even and for every $1\leq k\leq n$,

$$\sum_{i=1}^k d_i < k(k-1)+\sum_{i=k+1}^n\min(d_i,k)$$

(this is the strict Erdos Gallai theorem for the degrees of a graph).

Question: Can we find a qualitative bound on $r_i$ in terms of $n$? I would ideally like a bound of the form $n^c$ for some constant $c$ (that doesn't depend on $n$, $r_i$ or $d_i$). If such a bound doesn't exist, is there a counterexample or are there additional "mild" conditions on the $d_i$'s that can yield such a bound (for example, $d^2_{\mbox{max}}:=d^2_n\leq \frac{1}{2}\sum_{i=1}^nd_i$ gives a bound of $n^7$). ?

Notice that in the previous question, a bound of $n^{2n}$ was obtained but, sadly the posted argument doesn't seem to yield a better bound, unless I'm missing something obvious.

2 edited body

I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here:

For $i=1,2,\ldots,n$, let $d_i$ be a sequence of positive integers satisfying $d_1\geq\cdots\geq d_n$ and $r_i$ a sequence of positive real numbers with $2\leq d_i\leq n-1$ that satisfy the following system of equations:

$$d_i=\sum_{j=1}^n \frac{r_ir_j}{1+r_ir_j}$$

for each $i=1,2\ldots,n$.

I have additional constraints that $d_1+\cdots+d_n$ is even and for every $1\leq k\leq n$,

$$\sum_{i=1}^k d_i < k(k-1)+\sum_{i=k+1}^n\min(d_i,k)$$

(this is the strict Erdos Gallai theorem for the degrees of a graph).

Question: Can we find a qualitative bound on $r_i$ in terms of $n$? I would ideally like a bound of the form $n^c$ for some constant $c$ (that doesn't depend on $n$, $r_i$ or $d_i$). If such a bound doesn't exist, is there a counterexample or are there additional "mild" conditions on the $d_i$'s that can yield such a bound (for example, $d^2_{\mbox{max}}:=d^2_n\leq \frac{1}{2}\sum_{i=1}^nd_i$ gives a bound of $n^7$). ?

Notice that in the previous question, a bound of $n^{2n}$ was obtained but, sadly the posted argument doesn't seem to yield a better bound, unless I'm missing something obvious.

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# System of Equations Upper Bound

I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here:

For $i=1,2,\ldots,n$, let $d_i$ be a sequence of positive integers satisfying $d_1\geq\cdots\geq d_n$ and $r_i$ a sequence of positive real numbers with $2\leq d_i\leq n-1$ that satisfy the following system of equations:

$$d_i=\sum_{j=1}^n \frac{r_ir_j}{1+r_ir_j}$$

for each $i=1,2\ldots,n$.

I have additional constraints that $d_1+\cdots+d_n$ is even and for every $1\leq k\leq n$,

$$\sum_{i=1}^k d_i < k(k-1)+\sum_{i=k+1}^n\min(d_i,k)$$

(this is the strict Erdos Gallai theorem for the degrees of a graph).

Question: Can we find a qualitative bound on $r_i$ in terms of $n$? I would ideally like a bound of the form $n^c$ for some constant $c$ (that doesn't depend on $n$, $r_i$ or $d_i$). If such a bound doesn't exist, is there a counterexample or are there additional "mild" conditions on the $d_i$'s that can yield such a bound (for example, $d^2_{\mbox{max}}:=d^2_n\leq \frac{1}{2}\sum_{i=1}^nd_i$ gives a bound of $n^7$). ?

Notice that in the previous question, a bound of $n^{2n}$ was obtained but, sadly the posted argument doesn't seem to yield a better bound, unless I'm missing something obvious.